proof that contrapositive statement is logically equivalent to original statement

You can see that the contrapositive of an implication is true by consideringthe following:

The statement $p\Rightarrow q$ is logically equivalent to $\mathrm{\neg }p\vee q$ which can also be written as $\overline{p}\vee q$.

By the same token, the contrapositive statement $\overline{q}\Rightarrow \overline{p}$ is logically equivalent to $\mathrm{\neg }\overline{q}\vee \overline{p}$ which, using double negation on $q$, becomes $q\vee \overline{p}$.

This, of course, is the same logical statement.