Converse of Conditional is Contrapositive of Inverse

Theorem
Let $p \implies q$ be a conditional.

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

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


 * $q \implies p$

The inverse of $p \implies q$ is:


 * $\neg p \implies \neg q$

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


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

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