Paragraph

We want to start with one of the statements, and transform it into the other through a sequence of logically equivalent statements. Start with \(\neg(P \imp Q)\text{.}\) We can rewrite the implication as a disjunction this is logically equivalent to

\begin{equation*} \neg(\neg P \vee Q)\text{.} \end{equation*}

Now apply DeMorgan's law to get

\begin{equation*} \neg\neg P \wedge \neg Q\text{.} \end{equation*}

Finally, use double negation to arrive at \(P \wedge \neg Q\)

in-context