((\lnot A \to B) \land (\lnot A \to \lnot B)) \to A ("if not-A implies both B and its negation not-B, then not-A must be false, then A must be true"), which is the principle known as reductio ad absurdum.
((A \to B) \land (B \to C)) \to (A \to C) ("if A implies B and B implies C, then A implies C"), which is the principle known as syllogism.