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.