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https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_4 | [1152, 1] | [1183, 9] | simp only [isFreeIn] | case h1.h1.h4
P : Formula
u v : VarName
⊢ ¬isFreeIn u (forall_ u P.not_).not_ | case h1.h1.h4
P : Formula
u v : VarName
⊢ ¬(¬True ∧ isFreeIn u P) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_4 | [1152, 1] | [1183, 9] | simp | case h1.h1.h4
P : Formula
u v : VarName
⊢ ¬(¬True ∧ isFreeIn u P) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_4 | [1152, 1] | [1183, 9] | simp | case h1.h2
P : Formula
u v : VarName
⊢ ∀ H ∈ {exists_ u (forall_ v P)}, ¬isFreeIn v H | case h1.h2
P : Formula
u v : VarName
⊢ ¬isFreeIn v (exists_ u (forall_ v P)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_4 | [1152, 1] | [1183, 9] | simp only [def_exists_] | case h1.h2
P : Formula
u v : VarName
⊢ ¬isFreeIn v (exists_ u (forall_ v P)) | case h1.h2
P : Formula
u v : VarName
⊢ ¬isFreeIn v (forall_ u (forall_ v P).not_).not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_4 | [1152, 1] | [1183, 9] | simp only [isFreeIn] | case h1.h2
P : Formula
u v : VarName
⊢ ¬isFreeIn v (forall_ u (forall_ v P).not_).not_ | case h1.h2
P : Formula
u v : VarName
⊢ ¬(¬v = u ∧ ¬True ∧ isFreeIn v P) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_4 | [1152, 1] | [1183, 9] | simp | case h1.h2
P : Formula
u v : VarName
⊢ ¬(¬v = u ∧ ¬True ∧ isFreeIn v P) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_5 | [1186, 1] | [1200, 24] | apply IsDeduct.mp_ ((forall_ v P).iff_ P) | P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsProof ((forall_ v (P.iff_ Q)).imp_ (P.iff_ (forall_ v Q))) | case a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅ (((forall_ v P).iff_ P).imp_ ((forall_ v (P.iff_ Q)).imp_ (P.iff_ (forall_ v Q))))
case a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅ ((forall_ v P).iff_ P) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_5 | [1186, 1] | [1200, 24] | apply IsDeduct.mp_ ((forall_ v (P.iff_ Q)).imp_ ((forall_ v P).iff_ (forall_ v Q))) | case a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅ (((forall_ v P).iff_ P).imp_ ((forall_ v (P.iff_ Q)).imp_ (P.iff_ (forall_ v Q)))) | case a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅
(((forall_ v (P.iff_ Q)).imp_ ((forall_ v P).iff_ (forall_ v Q))).imp_
(((forall_ v P).iff_ P).imp_ ((forall_ v (P.iff_ Q)).imp_ (P.iff_ (forall_ v Q)))))
case a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅ ((forall_ v (P.iff_ Q)).imp_ ((forall_ v P).iff_ (forall_ v Q))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_5 | [1186, 1] | [1200, 24] | simp only [def_iff_] | case a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅
(((forall_ v (P.iff_ Q)).imp_ ((forall_ v P).iff_ (forall_ v Q))).imp_
(((forall_ v P).iff_ P).imp_ ((forall_ v (P.iff_ Q)).imp_ (P.iff_ (forall_ v Q))))) | case a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅
(((forall_ v ((P.imp_ Q).and_ (Q.imp_ P))).imp_
(((forall_ v P).imp_ (forall_ v Q)).and_ ((forall_ v Q).imp_ (forall_ v P)))).imp_
((((forall_ v P).imp_ P).and_ (P.imp_ (forall_ v P))).imp_
((forall_ v ((P.imp_ Q).and_ (Q.imp_ P))).imp_ ((P.imp_ (forall_ v Q)).and_ ((forall_ v Q).imp_ P))))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_5 | [1186, 1] | [1200, 24] | simp only [def_and_] | case a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅
(((forall_ v ((P.imp_ Q).and_ (Q.imp_ P))).imp_
(((forall_ v P).imp_ (forall_ v Q)).and_ ((forall_ v Q).imp_ (forall_ v P)))).imp_
((((forall_ v P).imp_ P).and_ (P.imp_ (forall_ v P))).imp_
((forall_ v ((P.imp_ Q).and_ (Q.imp_ P))).imp_ ((P.imp_ (forall_ v Q)).and_ ((forall_ v Q).imp_ P))))) | case a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅
(((forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_).imp_
(((forall_ v P).imp_ (forall_ v Q)).imp_ ((forall_ v Q).imp_ (forall_ v P)).not_).not_).imp_
((((forall_ v P).imp_ P).imp_ (P.imp_ (forall_ v P)).not_).not_.imp_
((forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_).imp_
((P.imp_ (forall_ v Q)).imp_ ((forall_ v Q).imp_ P).not_).not_))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_5 | [1186, 1] | [1200, 24] | SC | case a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅
(((forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_).imp_
(((forall_ v P).imp_ (forall_ v Q)).imp_ ((forall_ v Q).imp_ (forall_ v P)).not_).not_).imp_
((((forall_ v P).imp_ P).imp_ (P.imp_ (forall_ v P)).not_).not_.imp_
((forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_).imp_
((P.imp_ (forall_ v Q)).imp_ ((forall_ v Q).imp_ P).not_).not_))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_5 | [1186, 1] | [1200, 24] | exact T_18_1 P Q v | case a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅ ((forall_ v (P.iff_ Q)).imp_ ((forall_ v P).iff_ (forall_ v Q))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_5 | [1186, 1] | [1200, 24] | exact T_19_1 P v h1 | case a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅ ((forall_ v P).iff_ P) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | apply deduction_theorem | P Q : Formula
v : VarName
⊢ IsProof ((forall_ v (P.iff_ Q)).imp_ ((exists_ v P).imp_ (exists_ v Q))) | case h1
P Q : Formula
v : VarName
⊢ IsDeduct (∅ ∪ {forall_ v (P.iff_ Q)}) ((exists_ v P).imp_ (exists_ v Q)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | apply deduction_theorem | case h1
P Q : Formula
v : VarName
⊢ IsDeduct (∅ ∪ {forall_ v (P.iff_ Q)}) ((exists_ v P).imp_ (exists_ v Q)) | case h1.h1
P Q : Formula
v : VarName
⊢ IsDeduct (∅ ∪ {forall_ v (P.iff_ Q)} ∪ {exists_ v P}) (exists_ v Q) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | simp | case h1.h1
P Q : Formula
v : VarName
⊢ IsDeduct (∅ ∪ {forall_ v (P.iff_ Q)} ∪ {exists_ v P}) (exists_ v Q) | case h1.h1
P Q : Formula
v : VarName
⊢ IsDeduct {exists_ v P, forall_ v (P.iff_ Q)} (exists_ v Q) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | apply rule_C P (exists_ v Q) v {exists_ v P, forall_ v (P.iff_ Q)} | case h1.h1
P Q : Formula
v : VarName
⊢ IsDeduct {exists_ v P, forall_ v (P.iff_ Q)} (exists_ v Q) | case h1.h1.h1
P Q : Formula
v : VarName
⊢ IsDeduct {exists_ v P, forall_ v (P.iff_ Q)} (exists_ v P)
case h1.h1.h2
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v P, forall_ v (P.iff_ Q)} ∪ {P}) (exists_ v Q)
case h1.h1.h3
P Q : Formula
v : VarName
⊢ ∀ H ∈ {exists_ v P, forall_ v (P.iff_ Q)}, ¬isFreeIn v H
case h1.h1.h4
P Q : Formula
v : VarName
⊢ ¬isFreeIn v (exists_ v Q) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | apply IsDeduct.assume_ | case h1.h1.h1
P Q : Formula
v : VarName
⊢ IsDeduct {exists_ v P, forall_ v (P.iff_ Q)} (exists_ v P) | case h1.h1.h1.a
P Q : Formula
v : VarName
⊢ exists_ v P ∈ {exists_ v P, forall_ v (P.iff_ Q)} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | simp | case h1.h1.h1.a
P Q : Formula
v : VarName
⊢ exists_ v P ∈ {exists_ v P, forall_ v (P.iff_ Q)} | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | apply exists_intro Q v v | case h1.h1.h2
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v P, forall_ v (P.iff_ Q)} ∪ {P}) (exists_ v Q) | case h1.h1.h2.h1
P Q : Formula
v : VarName
⊢ fastAdmits v v Q
case h1.h1.h2.h2
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v P, forall_ v (P.iff_ Q)} ∪ {P}) (fastReplaceFree v v Q) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | apply fastAdmits_self | case h1.h1.h2.h1
P Q : Formula
v : VarName
⊢ fastAdmits v v Q | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | simp only [fastReplaceFree_self] | case h1.h1.h2.h2
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v P, forall_ v (P.iff_ Q)} ∪ {P}) (fastReplaceFree v v Q) | case h1.h1.h2.h2
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v P, forall_ v (P.iff_ Q)} ∪ {P}) Q |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | apply IsDeduct.mp_ P | case h1.h1.h2.h2
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v P, forall_ v (P.iff_ Q)} ∪ {P}) Q | case h1.h1.h2.h2.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v P, forall_ v (P.iff_ Q)} ∪ {P}) (P.imp_ Q)
case h1.h1.h2.h2.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v P, forall_ v (P.iff_ Q)} ∪ {P}) P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | apply IsDeduct.mp_ (P.iff_ Q) | case h1.h1.h2.h2.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v P, forall_ v (P.iff_ Q)} ∪ {P}) (P.imp_ Q) | case h1.h1.h2.h2.a.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v P, forall_ v (P.iff_ Q)} ∪ {P}) ((P.iff_ Q).imp_ (P.imp_ Q))
case h1.h1.h2.h2.a.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v P, forall_ v (P.iff_ Q)} ∪ {P}) (P.iff_ Q) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | simp only [def_iff_] | case h1.h1.h2.h2.a.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v P, forall_ v (P.iff_ Q)} ∪ {P}) ((P.iff_ Q).imp_ (P.imp_ Q)) | case h1.h1.h2.h2.a.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} ∪ {P}) (((P.imp_ Q).and_ (Q.imp_ P)).imp_ (P.imp_ Q)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | simp only [def_and_] | case h1.h1.h2.h2.a.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} ∪ {P}) (((P.imp_ Q).and_ (Q.imp_ P)).imp_ (P.imp_ Q)) | case h1.h1.h2.h2.a.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v P, forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_} ∪ {P})
(((P.imp_ Q).imp_ (Q.imp_ P).not_).not_.imp_ (P.imp_ Q)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | SC | case h1.h1.h2.h2.a.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v P, forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_} ∪ {P})
(((P.imp_ Q).imp_ (Q.imp_ P).not_).not_.imp_ (P.imp_ Q)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | apply specId v | case h1.h1.h2.h2.a.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v P, forall_ v (P.iff_ Q)} ∪ {P}) (P.iff_ Q) | case h1.h1.h2.h2.a.a.h1
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v P, forall_ v (P.iff_ Q)} ∪ {P}) (forall_ v (P.iff_ Q)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | apply IsDeduct.assume_ | case h1.h1.h2.h2.a.a.h1
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v P, forall_ v (P.iff_ Q)} ∪ {P}) (forall_ v (P.iff_ Q)) | case h1.h1.h2.h2.a.a.h1.a
P Q : Formula
v : VarName
⊢ forall_ v (P.iff_ Q) ∈ {exists_ v P, forall_ v (P.iff_ Q)} ∪ {P} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | simp | case h1.h1.h2.h2.a.a.h1.a
P Q : Formula
v : VarName
⊢ forall_ v (P.iff_ Q) ∈ {exists_ v P, forall_ v (P.iff_ Q)} ∪ {P} | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | apply IsDeduct.assume_ | case h1.h1.h2.h2.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v P, forall_ v (P.iff_ Q)} ∪ {P}) P | case h1.h1.h2.h2.a.a
P Q : Formula
v : VarName
⊢ P ∈ {exists_ v P, forall_ v (P.iff_ Q)} ∪ {P} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | simp | case h1.h1.h2.h2.a.a
P Q : Formula
v : VarName
⊢ P ∈ {exists_ v P, forall_ v (P.iff_ Q)} ∪ {P} | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | simp only [def_exists_] | case h1.h1.h3
P Q : Formula
v : VarName
⊢ ∀ H ∈ {exists_ v P, forall_ v (P.iff_ Q)}, ¬isFreeIn v H | case h1.h1.h3
P Q : Formula
v : VarName
⊢ ∀ H ∈ {(forall_ v P.not_).not_, forall_ v (P.iff_ Q)}, ¬isFreeIn v H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | simp | case h1.h1.h3
P Q : Formula
v : VarName
⊢ ∀ H ∈ {(forall_ v P.not_).not_, forall_ v (P.iff_ Q)}, ¬isFreeIn v H | case h1.h1.h3
P Q : Formula
v : VarName
⊢ ¬isFreeIn v (forall_ v P.not_).not_ ∧ ¬isFreeIn v (forall_ v (P.iff_ Q)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | simp only [isFreeIn] | case h1.h1.h3
P Q : Formula
v : VarName
⊢ ¬isFreeIn v (forall_ v P.not_).not_ ∧ ¬isFreeIn v (forall_ v (P.iff_ Q)) | case h1.h1.h3
P Q : Formula
v : VarName
⊢ ¬(¬True ∧ isFreeIn v P) ∧ ¬(¬True ∧ (isFreeIn v P ∨ isFreeIn v Q)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | simp | case h1.h1.h3
P Q : Formula
v : VarName
⊢ ¬(¬True ∧ isFreeIn v P) ∧ ¬(¬True ∧ (isFreeIn v P ∨ isFreeIn v Q)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | simp only [def_exists_] | case h1.h1.h4
P Q : Formula
v : VarName
⊢ ¬isFreeIn v (exists_ v Q) | case h1.h1.h4
P Q : Formula
v : VarName
⊢ ¬isFreeIn v (forall_ v Q.not_).not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | simp only [isFreeIn] | case h1.h1.h4
P Q : Formula
v : VarName
⊢ ¬isFreeIn v (forall_ v Q.not_).not_ | case h1.h1.h4
P Q : Formula
v : VarName
⊢ ¬(¬True ∧ isFreeIn v Q) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | simp | case h1.h1.h4
P Q : Formula
v : VarName
⊢ ¬(¬True ∧ isFreeIn v Q) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_right | [1240, 1] | [1262, 11] | apply deduction_theorem | P Q : Formula
v : VarName
⊢ IsProof ((forall_ v (P.iff_ Q)).imp_ ((exists_ v Q).imp_ (exists_ v P))) | case h1
P Q : Formula
v : VarName
⊢ IsDeduct (∅ ∪ {forall_ v (P.iff_ Q)}) ((exists_ v Q).imp_ (exists_ v P)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_right | [1240, 1] | [1262, 11] | simp | case h1
P Q : Formula
v : VarName
⊢ IsDeduct (∅ ∪ {forall_ v (P.iff_ Q)}) ((exists_ v Q).imp_ (exists_ v P)) | case h1
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v (P.iff_ Q)} ((exists_ v Q).imp_ (exists_ v P)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_right | [1240, 1] | [1262, 11] | apply IsDeduct.mp_ (forall_ v (Q.iff_ P)) | case h1
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v (P.iff_ Q)} ((exists_ v Q).imp_ (exists_ v P)) | case h1.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v (P.iff_ Q)} ((forall_ v (Q.iff_ P)).imp_ ((exists_ v Q).imp_ (exists_ v P)))
case h1.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v (P.iff_ Q)} (forall_ v (Q.iff_ P)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_right | [1240, 1] | [1262, 11] | apply proof_imp_deduct | case h1.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v (P.iff_ Q)} ((forall_ v (Q.iff_ P)).imp_ ((exists_ v Q).imp_ (exists_ v P))) | case h1.a.h1
P Q : Formula
v : VarName
⊢ IsProof ((forall_ v (Q.iff_ P)).imp_ ((exists_ v Q).imp_ (exists_ v P))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_right | [1240, 1] | [1262, 11] | apply T_19_6_left Q P v | case h1.a.h1
P Q : Formula
v : VarName
⊢ IsProof ((forall_ v (Q.iff_ P)).imp_ ((exists_ v Q).imp_ (exists_ v P))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_right | [1240, 1] | [1262, 11] | apply generalization | case h1.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v (P.iff_ Q)} (forall_ v (Q.iff_ P)) | case h1.a.h1
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v (P.iff_ Q)} (Q.iff_ P)
case h1.a.h2
P Q : Formula
v : VarName
⊢ ∀ H ∈ {forall_ v (P.iff_ Q)}, ¬isFreeIn v H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_right | [1240, 1] | [1262, 11] | apply IsDeduct.mp_ (P.iff_ Q) | case h1.a.h1
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v (P.iff_ Q)} (Q.iff_ P) | case h1.a.h1.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v (P.iff_ Q)} ((P.iff_ Q).imp_ (Q.iff_ P))
case h1.a.h1.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v (P.iff_ Q)} (P.iff_ Q) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_right | [1240, 1] | [1262, 11] | simp only [def_iff_] | case h1.a.h1.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v (P.iff_ Q)} ((P.iff_ Q).imp_ (Q.iff_ P)) | case h1.a.h1.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (((P.imp_ Q).and_ (Q.imp_ P)).imp_ ((Q.imp_ P).and_ (P.imp_ Q))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_right | [1240, 1] | [1262, 11] | simp only [def_and_] | case h1.a.h1.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (((P.imp_ Q).and_ (Q.imp_ P)).imp_ ((Q.imp_ P).and_ (P.imp_ Q))) | case h1.a.h1.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_}
(((P.imp_ Q).imp_ (Q.imp_ P).not_).not_.imp_ ((Q.imp_ P).imp_ (P.imp_ Q).not_).not_) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_right | [1240, 1] | [1262, 11] | SC | case h1.a.h1.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_}
(((P.imp_ Q).imp_ (Q.imp_ P).not_).not_.imp_ ((Q.imp_ P).imp_ (P.imp_ Q).not_).not_) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_right | [1240, 1] | [1262, 11] | apply specId v | case h1.a.h1.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v (P.iff_ Q)} (P.iff_ Q) | case h1.a.h1.a.h1
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v (P.iff_ Q)} (forall_ v (P.iff_ Q)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_right | [1240, 1] | [1262, 11] | apply IsDeduct.assume_ | case h1.a.h1.a.h1
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v (P.iff_ Q)} (forall_ v (P.iff_ Q)) | case h1.a.h1.a.h1.a
P Q : Formula
v : VarName
⊢ forall_ v (P.iff_ Q) ∈ {forall_ v (P.iff_ Q)} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_right | [1240, 1] | [1262, 11] | simp | case h1.a.h1.a.h1.a
P Q : Formula
v : VarName
⊢ forall_ v (P.iff_ Q) ∈ {forall_ v (P.iff_ Q)} | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_right | [1240, 1] | [1262, 11] | simp | case h1.a.h2
P Q : Formula
v : VarName
⊢ ∀ H ∈ {forall_ v (P.iff_ Q)}, ¬isFreeIn v H | case h1.a.h2
P Q : Formula
v : VarName
⊢ ¬isFreeIn v (forall_ v (P.iff_ Q)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_right | [1240, 1] | [1262, 11] | simp only [isFreeIn] | case h1.a.h2
P Q : Formula
v : VarName
⊢ ¬isFreeIn v (forall_ v (P.iff_ Q)) | case h1.a.h2
P Q : Formula
v : VarName
⊢ ¬(¬True ∧ (isFreeIn v P ∨ isFreeIn v Q)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_right | [1240, 1] | [1262, 11] | simp | case h1.a.h2
P Q : Formula
v : VarName
⊢ ¬(¬True ∧ (isFreeIn v P ∨ isFreeIn v Q)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6 | [1265, 1] | [1280, 22] | apply IsDeduct.mp_ ((forall_ v (P.iff_ Q)).imp_ ((exists_ v P).imp_ (exists_ v Q))) | P Q : Formula
v : VarName
⊢ IsProof ((forall_ v (P.iff_ Q)).imp_ ((exists_ v P).iff_ (exists_ v Q))) | case a
P Q : Formula
v : VarName
⊢ IsDeduct ∅
(((forall_ v (P.iff_ Q)).imp_ ((exists_ v P).imp_ (exists_ v Q))).imp_
((forall_ v (P.iff_ Q)).imp_ ((exists_ v P).iff_ (exists_ v Q))))
case a
P Q : Formula
v : VarName
⊢ IsDeduct ∅ ((forall_ v (P.iff_ Q)).imp_ ((exists_ v P).imp_ (exists_ v Q))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6 | [1265, 1] | [1280, 22] | apply IsDeduct.mp_ ((forall_ v (P.iff_ Q)).imp_ ((exists_ v Q).imp_ (exists_ v P))) | case a
P Q : Formula
v : VarName
⊢ IsDeduct ∅
(((forall_ v (P.iff_ Q)).imp_ ((exists_ v P).imp_ (exists_ v Q))).imp_
((forall_ v (P.iff_ Q)).imp_ ((exists_ v P).iff_ (exists_ v Q)))) | case a.a
P Q : Formula
v : VarName
⊢ IsDeduct ∅
(((forall_ v (P.iff_ Q)).imp_ ((exists_ v Q).imp_ (exists_ v P))).imp_
(((forall_ v (P.iff_ Q)).imp_ ((exists_ v P).imp_ (exists_ v Q))).imp_
((forall_ v (P.iff_ Q)).imp_ ((exists_ v P).iff_ (exists_ v Q)))))
case a.a
P Q : Formula
v : VarName
⊢ IsDeduct ∅ ((forall_ v (P.iff_ Q)).imp_ ((exists_ v Q).imp_ (exists_ v P))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6 | [1265, 1] | [1280, 22] | simp only [def_exists_] | case a.a
P Q : Formula
v : VarName
⊢ IsDeduct ∅
(((forall_ v (P.iff_ Q)).imp_ ((exists_ v Q).imp_ (exists_ v P))).imp_
(((forall_ v (P.iff_ Q)).imp_ ((exists_ v P).imp_ (exists_ v Q))).imp_
((forall_ v (P.iff_ Q)).imp_ ((exists_ v P).iff_ (exists_ v Q))))) | case a.a
P Q : Formula
v : VarName
⊢ IsDeduct ∅
(((forall_ v (P.iff_ Q)).imp_ ((forall_ v Q.not_).not_.imp_ (forall_ v P.not_).not_)).imp_
(((forall_ v (P.iff_ Q)).imp_ ((forall_ v P.not_).not_.imp_ (forall_ v Q.not_).not_)).imp_
((forall_ v (P.iff_ Q)).imp_ ((forall_ v P.not_).not_.iff_ (forall_ v Q.not_).not_)))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6 | [1265, 1] | [1280, 22] | simp only [def_iff_] | case a.a
P Q : Formula
v : VarName
⊢ IsDeduct ∅
(((forall_ v (P.iff_ Q)).imp_ ((forall_ v Q.not_).not_.imp_ (forall_ v P.not_).not_)).imp_
(((forall_ v (P.iff_ Q)).imp_ ((forall_ v P.not_).not_.imp_ (forall_ v Q.not_).not_)).imp_
((forall_ v (P.iff_ Q)).imp_ ((forall_ v P.not_).not_.iff_ (forall_ v Q.not_).not_)))) | case a.a
P Q : Formula
v : VarName
⊢ IsDeduct ∅
(((forall_ v ((P.imp_ Q).and_ (Q.imp_ P))).imp_ ((forall_ v Q.not_).not_.imp_ (forall_ v P.not_).not_)).imp_
(((forall_ v ((P.imp_ Q).and_ (Q.imp_ P))).imp_ ((forall_ v P.not_).not_.imp_ (forall_ v Q.not_).not_)).imp_
((forall_ v ((P.imp_ Q).and_ (Q.imp_ P))).imp_
(((forall_ v P.not_).not_.imp_ (forall_ v Q.not_).not_).and_
((forall_ v Q.not_).not_.imp_ (forall_ v P.not_).not_))))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6 | [1265, 1] | [1280, 22] | simp only [def_and_] | case a.a
P Q : Formula
v : VarName
⊢ IsDeduct ∅
(((forall_ v ((P.imp_ Q).and_ (Q.imp_ P))).imp_ ((forall_ v Q.not_).not_.imp_ (forall_ v P.not_).not_)).imp_
(((forall_ v ((P.imp_ Q).and_ (Q.imp_ P))).imp_ ((forall_ v P.not_).not_.imp_ (forall_ v Q.not_).not_)).imp_
((forall_ v ((P.imp_ Q).and_ (Q.imp_ P))).imp_
(((forall_ v P.not_).not_.imp_ (forall_ v Q.not_).not_).and_
((forall_ v Q.not_).not_.imp_ (forall_ v P.not_).not_))))) | case a.a
P Q : Formula
v : VarName
⊢ IsDeduct ∅
(((forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_).imp_
((forall_ v Q.not_).not_.imp_ (forall_ v P.not_).not_)).imp_
(((forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_).imp_
((forall_ v P.not_).not_.imp_ (forall_ v Q.not_).not_)).imp_
((forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_).imp_
(((forall_ v P.not_).not_.imp_ (forall_ v Q.not_).not_).imp_
((forall_ v Q.not_).not_.imp_ (forall_ v P.not_).not_).not_).not_))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6 | [1265, 1] | [1280, 22] | SC | case a.a
P Q : Formula
v : VarName
⊢ IsDeduct ∅
(((forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_).imp_
((forall_ v Q.not_).not_.imp_ (forall_ v P.not_).not_)).imp_
(((forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_).imp_
((forall_ v P.not_).not_.imp_ (forall_ v Q.not_).not_)).imp_
((forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_).imp_
(((forall_ v P.not_).not_.imp_ (forall_ v Q.not_).not_).imp_
((forall_ v Q.not_).not_.imp_ (forall_ v P.not_).not_).not_).not_))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6 | [1265, 1] | [1280, 22] | apply T_19_6_right | case a.a
P Q : Formula
v : VarName
⊢ IsDeduct ∅ ((forall_ v (P.iff_ Q)).imp_ ((exists_ v Q).imp_ (exists_ v P))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6 | [1265, 1] | [1280, 22] | apply T_19_6_left | case a
P Q : Formula
v : VarName
⊢ IsDeduct ∅ ((forall_ v (P.iff_ Q)).imp_ ((exists_ v P).imp_ (exists_ v Q))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_left | [1283, 1] | [1301, 26] | apply C_18_4 (forall_ v P) P ((forall_ v (P.imp_ Q)).imp_ ((forall_ v P).imp_ (forall_ v Q))) | P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsProof ((forall_ v (P.imp_ Q)).imp_ (P.imp_ (forall_ v Q))) | case h1
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsReplOfFormulaInFormula (forall_ v P) P ((forall_ v (P.imp_ Q)).imp_ ((forall_ v P).imp_ (forall_ v Q)))
((forall_ v (P.imp_ Q)).imp_ (P.imp_ (forall_ v Q)))
case h2
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsProof ((forall_ v P).iff_ P)
case h3
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅ ((forall_ v (P.imp_ Q)).imp_ ((forall_ v P).imp_ (forall_ v Q))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_left | [1283, 1] | [1301, 26] | apply IsReplOfFormulaInFormula.imp_ | case h1
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsReplOfFormulaInFormula (forall_ v P) P ((forall_ v (P.imp_ Q)).imp_ ((forall_ v P).imp_ (forall_ v Q)))
((forall_ v (P.imp_ Q)).imp_ (P.imp_ (forall_ v Q))) | case h1.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsReplOfFormulaInFormula (forall_ v P) P (forall_ v (P.imp_ Q)) (forall_ v (P.imp_ Q))
case h1.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsReplOfFormulaInFormula (forall_ v P) P ((forall_ v P).imp_ (forall_ v Q)) (P.imp_ (forall_ v Q)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_left | [1283, 1] | [1301, 26] | apply IsReplOfFormulaInFormula.same_ | case h1.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsReplOfFormulaInFormula (forall_ v P) P (forall_ v (P.imp_ Q)) (forall_ v (P.imp_ Q)) | case h1.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ forall_ v (P.imp_ Q) = forall_ v (P.imp_ Q) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_left | [1283, 1] | [1301, 26] | rfl | case h1.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ forall_ v (P.imp_ Q) = forall_ v (P.imp_ Q) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_left | [1283, 1] | [1301, 26] | apply IsReplOfFormulaInFormula.imp_ | case h1.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsReplOfFormulaInFormula (forall_ v P) P ((forall_ v P).imp_ (forall_ v Q)) (P.imp_ (forall_ v Q)) | case h1.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsReplOfFormulaInFormula (forall_ v P) P (forall_ v P) P
case h1.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsReplOfFormulaInFormula (forall_ v P) P (forall_ v Q) (forall_ v Q) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_left | [1283, 1] | [1301, 26] | apply IsReplOfFormulaInFormula.diff_ | case h1.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsReplOfFormulaInFormula (forall_ v P) P (forall_ v P) P | case h1.a.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ forall_ v P = forall_ v P
case h1.a.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ P = P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_left | [1283, 1] | [1301, 26] | rfl | case h1.a.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ forall_ v P = forall_ v P | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_left | [1283, 1] | [1301, 26] | rfl | case h1.a.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ P = P | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_left | [1283, 1] | [1301, 26] | apply IsReplOfFormulaInFormula.same_ | case h1.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsReplOfFormulaInFormula (forall_ v P) P (forall_ v Q) (forall_ v Q) | case h1.a.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ forall_ v Q = forall_ v Q |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_left | [1283, 1] | [1301, 26] | rfl | case h1.a.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ forall_ v Q = forall_ v Q | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_left | [1283, 1] | [1301, 26] | exact T_19_1 P v h1 | case h2
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsProof ((forall_ v P).iff_ P) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_left | [1283, 1] | [1301, 26] | apply IsDeduct.axiom_ | case h3
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅ ((forall_ v (P.imp_ Q)).imp_ ((forall_ v P).imp_ (forall_ v Q))) | case h3.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsAxiom ((forall_ v (P.imp_ Q)).imp_ ((forall_ v P).imp_ (forall_ v Q))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_left | [1283, 1] | [1301, 26] | apply IsAxiom.pred_1_ | case h3.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsAxiom ((forall_ v (P.imp_ Q)).imp_ ((forall_ v P).imp_ (forall_ v Q))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_right | [1304, 1] | [1325, 13] | apply deduction_theorem | P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsProof ((P.imp_ (forall_ v Q)).imp_ (forall_ v (P.imp_ Q))) | case h1
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct (∅ ∪ {P.imp_ (forall_ v Q)}) (forall_ v (P.imp_ Q)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_right | [1304, 1] | [1325, 13] | simp | case h1
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct (∅ ∪ {P.imp_ (forall_ v Q)}) (forall_ v (P.imp_ Q)) | case h1
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct {P.imp_ (forall_ v Q)} (forall_ v (P.imp_ Q)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_right | [1304, 1] | [1325, 13] | apply generalization | case h1
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct {P.imp_ (forall_ v Q)} (forall_ v (P.imp_ Q)) | case h1.h1
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct {P.imp_ (forall_ v Q)} (P.imp_ Q)
case h1.h2
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ ∀ H ∈ {P.imp_ (forall_ v Q)}, ¬isFreeIn v H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_right | [1304, 1] | [1325, 13] | apply deduction_theorem | case h1.h1
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct {P.imp_ (forall_ v Q)} (P.imp_ Q) | case h1.h1.h1
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ({P.imp_ (forall_ v Q)} ∪ {P}) Q |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_right | [1304, 1] | [1325, 13] | apply specId v | case h1.h1.h1
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ({P.imp_ (forall_ v Q)} ∪ {P}) Q | case h1.h1.h1.h1
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ({P.imp_ (forall_ v Q)} ∪ {P}) (forall_ v Q) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_right | [1304, 1] | [1325, 13] | apply IsDeduct.mp_ P | case h1.h1.h1.h1
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ({P.imp_ (forall_ v Q)} ∪ {P}) (forall_ v Q) | case h1.h1.h1.h1.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ({P.imp_ (forall_ v Q)} ∪ {P}) (P.imp_ (forall_ v Q))
case h1.h1.h1.h1.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ({P.imp_ (forall_ v Q)} ∪ {P}) P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_right | [1304, 1] | [1325, 13] | apply IsDeduct.assume_ | case h1.h1.h1.h1.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ({P.imp_ (forall_ v Q)} ∪ {P}) (P.imp_ (forall_ v Q)) | case h1.h1.h1.h1.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ P.imp_ (forall_ v Q) ∈ {P.imp_ (forall_ v Q)} ∪ {P} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_right | [1304, 1] | [1325, 13] | simp | case h1.h1.h1.h1.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ P.imp_ (forall_ v Q) ∈ {P.imp_ (forall_ v Q)} ∪ {P} | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_right | [1304, 1] | [1325, 13] | apply IsDeduct.assume_ | case h1.h1.h1.h1.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ({P.imp_ (forall_ v Q)} ∪ {P}) P | case h1.h1.h1.h1.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ P ∈ {P.imp_ (forall_ v Q)} ∪ {P} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_right | [1304, 1] | [1325, 13] | simp | case h1.h1.h1.h1.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ P ∈ {P.imp_ (forall_ v Q)} ∪ {P} | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_right | [1304, 1] | [1325, 13] | intro H a1 | case h1.h2
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ ∀ H ∈ {P.imp_ (forall_ v Q)}, ¬isFreeIn v H | case h1.h2
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
H : Formula
a1 : H ∈ {P.imp_ (forall_ v Q)}
⊢ ¬isFreeIn v H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_right | [1304, 1] | [1325, 13] | simp at a1 | case h1.h2
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
H : Formula
a1 : H ∈ {P.imp_ (forall_ v Q)}
⊢ ¬isFreeIn v H | case h1.h2
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
H : Formula
a1 : H = P.imp_ (forall_ v Q)
⊢ ¬isFreeIn v H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_right | [1304, 1] | [1325, 13] | subst a1 | case h1.h2
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
H : Formula
a1 : H = P.imp_ (forall_ v Q)
⊢ ¬isFreeIn v H | case h1.h2
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ ¬isFreeIn v (P.imp_ (forall_ v Q)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_right | [1304, 1] | [1325, 13] | simp only [isFreeIn] | case h1.h2
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ ¬isFreeIn v (P.imp_ (forall_ v Q)) | case h1.h2
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ ¬(isFreeIn v P ∨ ¬True ∧ isFreeIn v Q) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_right | [1304, 1] | [1325, 13] | simp | case h1.h2
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ ¬(isFreeIn v P ∨ ¬True ∧ isFreeIn v Q) | case h1.h2
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ ¬isFreeIn v P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_right | [1304, 1] | [1325, 13] | exact h1 | case h1.h2
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ ¬isFreeIn v P | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21 | [1328, 1] | [1342, 35] | apply IsDeduct.mp_ ((forall_ v (P.imp_ Q)).imp_ (P.imp_ (forall_ v Q))) | P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsProof ((forall_ v (P.imp_ Q)).iff_ (P.imp_ (forall_ v Q))) | case a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅
(((forall_ v (P.imp_ Q)).imp_ (P.imp_ (forall_ v Q))).imp_ ((forall_ v (P.imp_ Q)).iff_ (P.imp_ (forall_ v Q))))
case a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅ ((forall_ v (P.imp_ Q)).imp_ (P.imp_ (forall_ v Q))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21 | [1328, 1] | [1342, 35] | apply IsDeduct.mp_ ((P.imp_ (forall_ v Q)).imp_ (forall_ v (P.imp_ Q))) | case a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅
(((forall_ v (P.imp_ Q)).imp_ (P.imp_ (forall_ v Q))).imp_ ((forall_ v (P.imp_ Q)).iff_ (P.imp_ (forall_ v Q)))) | case a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅
(((P.imp_ (forall_ v Q)).imp_ (forall_ v (P.imp_ Q))).imp_
(((forall_ v (P.imp_ Q)).imp_ (P.imp_ (forall_ v Q))).imp_ ((forall_ v (P.imp_ Q)).iff_ (P.imp_ (forall_ v Q)))))
case a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅ ((P.imp_ (forall_ v Q)).imp_ (forall_ v (P.imp_ Q))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21 | [1328, 1] | [1342, 35] | simp only [def_iff_] | case a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅
(((P.imp_ (forall_ v Q)).imp_ (forall_ v (P.imp_ Q))).imp_
(((forall_ v (P.imp_ Q)).imp_ (P.imp_ (forall_ v Q))).imp_ ((forall_ v (P.imp_ Q)).iff_ (P.imp_ (forall_ v Q))))) | case a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅
(((P.imp_ (forall_ v Q)).imp_ (forall_ v (P.imp_ Q))).imp_
(((forall_ v (P.imp_ Q)).imp_ (P.imp_ (forall_ v Q))).imp_
(((forall_ v (P.imp_ Q)).imp_ (P.imp_ (forall_ v Q))).and_
((P.imp_ (forall_ v Q)).imp_ (forall_ v (P.imp_ Q)))))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21 | [1328, 1] | [1342, 35] | simp only [def_and_] | case a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅
(((P.imp_ (forall_ v Q)).imp_ (forall_ v (P.imp_ Q))).imp_
(((forall_ v (P.imp_ Q)).imp_ (P.imp_ (forall_ v Q))).imp_
(((forall_ v (P.imp_ Q)).imp_ (P.imp_ (forall_ v Q))).and_
((P.imp_ (forall_ v Q)).imp_ (forall_ v (P.imp_ Q)))))) | case a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅
(((P.imp_ (forall_ v Q)).imp_ (forall_ v (P.imp_ Q))).imp_
(((forall_ v (P.imp_ Q)).imp_ (P.imp_ (forall_ v Q))).imp_
(((forall_ v (P.imp_ Q)).imp_ (P.imp_ (forall_ v Q))).imp_
((P.imp_ (forall_ v Q)).imp_ (forall_ v (P.imp_ Q))).not_).not_)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21 | [1328, 1] | [1342, 35] | SC | case a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅
(((P.imp_ (forall_ v Q)).imp_ (forall_ v (P.imp_ Q))).imp_
(((forall_ v (P.imp_ Q)).imp_ (P.imp_ (forall_ v Q))).imp_
(((forall_ v (P.imp_ Q)).imp_ (P.imp_ (forall_ v Q))).imp_
((P.imp_ (forall_ v Q)).imp_ (forall_ v (P.imp_ Q))).not_).not_)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21 | [1328, 1] | [1342, 35] | exact T_19_TS_21_right P Q v h1 | case a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅ ((P.imp_ (forall_ v Q)).imp_ (forall_ v (P.imp_ Q))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21 | [1328, 1] | [1342, 35] | exact T_19_TS_21_left P Q v h1 | case a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅ ((forall_ v (P.imp_ Q)).imp_ (P.imp_ (forall_ v Q))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_1 | [1345, 1] | [1370, 15] | apply generalization | x y : VarName
⊢ IsProof (forall_ x (forall_ y ((eq_ x y).imp_ (eq_ y x)))) | case h1
x y : VarName
⊢ IsDeduct ∅ (forall_ y ((eq_ x y).imp_ (eq_ y x)))
case h2
x y : VarName
⊢ ∀ H ∈ ∅, ¬isFreeIn x H |
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