Need help with Hilbert-style proof (Given ¬q and (¬p⇒(¬q⇒¬r)), prove (r⇒p))

Question

Given $\neg q$ and $(\neg p⇒(¬q⇒¬r))$, prove $(r⇒p)$. I unfortunately can't figure out how to begin this proof. Do any of you have any idea how to begin this proof? I'd appreciate any help anyone can provide.

Thank you!

Answer 1

Note that $\neg p \to (\neg q \to \neg r)$ is logically equivalent to $( \neg p \land q) \lor (r \to p)$. The result is trivial now given $\neg q$.

Answer 2

That depends on how you are supposed to do this. What I would do is:

1. Use Permutation to get ${\neg} q \Rightarrow ({\neg}p \Rightarrow {\neg}r)$
2. Detach to get ${\neg}p \Rightarrow {\neg}r$
3. Use Contraposition to get $r \Rightarrow p$