Prove the tautology (⋀( → ¬)) → ¬

Question

I must prove this tautology using logical equivalences but I can't quite figure it out. I know it has something to do with the fact that not p and p have opposite truth values at all times. Any help would be appreciated.

Answer 1

Rewrite $p\to\neg q$ as $q\to\neg p$ (both are equivalent to $\neg p\lor\neg q$).

Answer 2

Use the fact that $p \rightarrow q \Leftrightarrow \neg p \lor q$

Applied to your statement:

$(q \land (p \rightarrow \neg )) \rightarrow \neg p \Leftrightarrow$

$\neg (q \land (p \rightarrow \neg )) \lor \neg p \Leftrightarrow$

$\neg q \lor \neg (p \rightarrow \neg ) \lor \neg p \Leftrightarrow$

$\neg q \lor \neg (\neg p \lor \neg ) \lor \neg p \Leftrightarrow$

$\neg q \lor (p \land q) \lor \neg p \Leftrightarrow$

$((\neg q \lor p) \land (\neg q \lor q)) \lor \neg p \Leftrightarrow$

$((\neg q \lor p) \land \top) \lor \neg p \Leftrightarrow$

$(\neg q \lor p) \lor \neg p \Leftrightarrow$

$\neg q \lor p \lor \neg p$

... and now you're almost there ... do you see it?