A formal proof that $ (P \Rightarrow Q) $ is equivalent to its contrapositive $ (\neg Q \Rightarrow \neg P) $.

Question

I know how to show this via truth tables, but I’m confused over the formal proof.

Wikipedia tells me that:

$ (P \Rightarrow Q) \equiv (\neg P \lor Q) \equiv (Q \lor \neg P) \equiv (\neg Q \Rightarrow \neg P) $.

I don’t understand how we get from $ (P \Rightarrow Q) $ to $ (\neg P \lor Q) $. I also don’t understand how $ (Q \lor \neg P) $ implies $ (\neg Q \Rightarrow \neg P) $.

Answer 1

$ (P \Rightarrow Q) \equiv (\neg P \lor Q) \equiv (Q \lor \neg P) \equiv (\neg \neg Q \lor \neg P) \equiv (\neg Q \Rightarrow \neg P) $.