Inverse of Conditional is Converse of Contrapositive

Theorem

Let $p \implies q$ be a conditional.

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

Proof

The inverse of $p \implies q$ is:

$\neg p \implies \neg q$

The contrapositive of $p \implies q$ is:

$\neg q \implies \neg p$

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

$\neg p \implies \neg q$

The two are seen to be equal.

$\blacksquare$