I've started to work through Applied Mathematics for Database Professionals and have been stuck on one of the exercises for two days. I've been able to prove the expression: $\left( P\Rightarrow Q \right) \Leftrightarrow \left( \neg Q \Rightarrow \neg P \right)$ is true using a truth table but can't make the leap using rewrite rules.
Starting with the implication: $ P \Rightarrow Q$ Rewrite implication into disjunction: $\neg P \vee Q $ By commutativity the expression becomes: $ Q \vee \neg P $ Via double negation: $ \neg \neg Q \vee \neg P $ This is where I get stuck. The answer in the book shows rewriting the expression back into an implication: $ \neg Q \Rightarrow \neg P $ I understand where the $ \neg P $ comes from, it's from the initial rewrite rule. My confusion is, how is the $ \neg Q $ derived?