I am currently working through Velleman's book How To Prove It and was asked to prove the following
$(P \leftrightarrow Q) \equiv (P \wedge Q) \vee (\neg P \wedge \neg Q)$
This is my work thus far
$(P \to Q) \wedge (Q \to P)$
$(\neg P \vee Q) \wedge (\neg Q \vee P)$ (since $(P \to Q) \equiv (\neg P \vee Q)$)
$\neg[\neg(\neg P \vee Q) \vee \neg (\neg Q \vee P)]$ (Demorgan's Law)
$\neg [(P \wedge \neg Q) \vee (Q \wedge \neg P)]$ (Demorgan's Law)
At this point I am little unsure how to proceed.
Here are a few things I've tried or considered thus far:
I thought that I could perhaps switch some of the terms in step 3 using the law of associativity however the $\neg$ on the outside of the two terms prevents me from doing so (constructing a truth table for $\neg (\neg P \vee Q) \vee (\neg Q \vee P)$ and $\neg (\neg P \vee \neg Q) \vee \neg (P \vee Q)$ for sanity purposes)
I can't seem to apply the law of distribution since the corresponding terms are negated.
Applying demorgans law to one of the terms individually on step 2 or 3 doesnt seem to get me very far either.
Did I perhaps skip something? Am I even on the right track? Any help is appreciated