What does it mean to prove if $p$ then $q$? Does that mean to prove $p\rightarrow q$ is a true statement? Then since when $p$ is false, $p\rightarrow q$ is vacuously true, do I only have to prove $q$ is true when $p$ is true?

In classical logic, $p \implies q$ means only that it is false that both $p$ is true and $q$ is false. 

$p \implies q \space \space \equiv \space\space \neg (p \land \neg q)$

We can prove $p\implies q$ by either:


*

*Assuming $p$ is true, and then proving that $q$ must also be true.

*Assuming $q$ is false, and then proving that $p$ must also be false.

*Assuming $p$ is true and $q$ is false, and then obtainimg a contradiction of the forms $r\land \neg r$ or $r \iff \neg r$ 

*Proving $p$ is false. (Then there is no need to prove anything about $q$.)

*Proving $q$ is true. (Then there is no need to prove anything about $p$.)
