Converse of Conditional is Inverse of Contrapositive

Theorem

Let $p \implies q$ be a conditional.

Then the converse of $p \implies q$ is the inverse of its contrapositive.

Proof

The converse of $p \implies q$ is:

$q \implies p$

The contrapositive of $p \implies q$ is:

$\neg q \implies \neg p$

The inverse of $\neg q \implies \neg p$ is:

$\neg \neg q \implies \neg \neg p$

By Double Negation, the two are seen to be equal.

$\blacksquare$