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\begin{equation*} \begin{split}x \leftrightarrow (p \rightarrow q) &\Leftrightarrow x \leftrightarrow (p \rightarrow q)\\&\Leftrightarrow (x \rightarrow (p \rightarrow q)) \land ((p \rightarrow q) \rightarrow x)\\&\Leftrightarrow (x \rightarrow (\lnot p \lor q)) \land ((\lnot p \lor q) \rightarrow x)\\&\Leftrightarrow (\lnot x \lor (\lnot p \lor q)) \land (\lnot (\lnot p \lor q) \lor x)\\&\Leftrightarrow (\lnot x \lor \lnot p \lor q) \land ((\lnot\lnot p \land \lnot q) \lor x)\\&\Leftrightarrow (\lnot x \lor \lnot p \lor q) \land ((p \land \lnot q) \lor x)\\&\Leftrightarrow (\lnot x \lor \lnot p \lor q) \land ((p \lor x) \land (\lnot q \lor x))\\&\Leftrightarrow (\lnot x \lor \lnot p \lor q) \land (p \lor x) \land (\lnot q \lor x)\\\end{split}\end{equation*}
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