Inverse of Conditional is Contrapositive of Converse

Theorem

Let $p \implies q$ be a conditional.

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

Proof

The inverse of $p \implies q$ is:

$\neg p \implies \neg q$

The converse of $p \implies q$ is:

$q \implies p$

The contrapositive of $q \implies p$ is:

$\neg p \implies \neg q$

The two are seen to be equal.

$\blacksquare$