Logical equivalences for $P \implies Q$ and $\neg Q \implies \neg P$ This is probably quite a basic question, but it's something I'm having trouble wrapping my head around.
I came across a proof which employed the following strategy
The goal was to prove 
$$P \implies Q$$
And the strategy used was to prove the following instead:
$$\neg Q \implies \neg P$$
If possible, could someone provide some intuition as to why these two statements would be equivalent? 

In classical logic this boils down to the definition of $\implies$. You can then check this equivalence by testing all possible combinations of truth values for $P$ and $Q$. This principle is called contraposition. 
In intuitionistic logic, you only get that $P\implies Q$ implies $\lnot Q\implies \lnot P$, but not the converse.
Contraposition is often a convenient way of starting a proof, in particular if you don't know where you want to go with your proof. But many such proofs can be in fact reformulated to be direct proofs.
