Determine the truth value of: $((p \to \neg q)\space\land (\neg r \lor q) \land r) \to \neg p$ without truth table

Question

Determine the truth value of: $$((p \to \neg q)\space\land (\neg r \lor q) \land r) \to \neg p$$

I can determinate it easily with truth tables (it's a tautology), but i want to do it without the table.

Any hints?

Answer 1

Suppose $((p\Rightarrow \lnot q) \land (\lnot r\lor q)\land r)\Rightarrow\lnot p$ is false. Then $(p\Rightarrow \lnot q) \land (\lnot r\lor q)\land r$ is true and $\lnot p$ is false. Then $p$ must be true. Furthermore, $p\Rightarrow \lnot q$ is true which implies that $\lnot q$ must be true. So $q$ is false. $\lnot r\lor q$ is also true, so $\lnot r$ must be true. But then $r$ is false. This is impossible (since $r$ must be true for $(p\Rightarrow \lnot q) \land (\lnot r\lor q)\land r$ to be true), so $((p\Rightarrow \lnot q) \land (\lnot r\lor q)\land r)\Rightarrow\lnot p$ must actually be true, a tautology.