Question

Need help with a proof

Answer 1

Details to come

Answer 2

I am trying to prove the following statement\[( \neg B \rightarrow \neg A) \rightarrow (A \rightarrow B)\] It clearly is showing the relationship between an implication and its contrapositive. We are in a propositional logic system where lines in the proof can consist of either instances and an axiom (which will be given), hypotheses (none are given for this particular proof), a previously proven statement, or application of modus ponens. The axioms of our system are\[A \rightarrow (B \rightarrow A)\]\[(A \rightarrow (B \rightarrow C)) \rightarrow ((A \rightarrow B) \rightarrow (A \rightarrow C))\]\[(\neg B \rightarrow \neg A) \rightarrow ((\neg B \rightarrow A) \rightarrow B)\] I have using the first axiom and defining A and (A implies B) and B defined as (not B implies not A) which yields\[(A \rightarrow B) \rightarrow ((\neg B \rightarrow \neg A) \rightarrow (A \rightarrow B))\] If I can deduce A implies B, then modus ponens will apply and the proof is complete.