Inverse of Conditional is Contrapositive of Converse
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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.
$\blacksquare$