The result I wish to prove is
(A -> (B -> C)) -> (B -> (A -> C))
Firstly does this have a name? I've been calling it "Swapping Hypothesis".
Secondly, I'm trying to find a proof of this only using the axioms from the book "Introduction to Mathematical Logic" by Mendelson. He gives three propositional axioms, here are 2 of those axioms
(A -> (B -> A)
((A -> (B -> C)) -> ((A -> B) -> (A -> C))
The rule of proof is Given
A -> B then
The third axiom I do not believe is relevant.
I've already proved a couple of results, which may be of use:
(B -> C) -> ((A -> B) -> (A -> C))
- (Rule) Given
(A -> B)and
(B -> C)then
(A -> C)(Hypothetical Syllogism)
If you can do it without Deduction Theorem that would be great, but it's OK with Deduction also (as I think I can translate a proof with Deduction into a proof without fairly easily).
UPDATE Found answer here (T4):