I am trying to proof a few statements in the deductive system L, in propositional logic. The system contains 3 axioms (I, II, III below) and a few proven statements (1,2,3,4). In addition, the only inference rule is the modus ponens. In a book I found, the system differs from mine in the 3rd axiom. In the book, the 3rd axiom is (~b->~a)->((~b->a)->b) while my 3rd axiom is (~b->~a)->(a->b). I wanted to ask, is it possible, given my axioms, statements the MP rule and the deduction theorem, to prove the 3rd axiom from the book? Thank you.

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