Converse of Conditional is Contrapositive of Inverse

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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.

$\blacksquare$