$$\begin{align}
&\quad \exists x \forall y \Biggl[ \color{blue}{\neg \: \exists z \Bigl [ p (y, z) \land p (z, y) \Bigr ]} \; \equiv \; \color{green}{p(x,y)} \Biggr ] \\[1em]
\equiv&\quad \exists x \forall y \Biggl [ \color{blue}{\neg \: \exists z \Bigl [ p (y, z) \land p (z, y) \Bigr ]} \longleftrightarrow \color{green}{p(x,y)} \Biggr ] \\[1em]
\equiv&\quad \exists x \forall y \Biggl [ \Bigl [\color{blue}{\neg \exists z \bigl [ p (y, z) \land p (z, y) \bigr ]} \land \color{green}{p(x,y)} \Bigr ] \lor \Bigl [ \color{red}{\neg} \color{blue}{\neg \exists z \bigl [ p (y, z) \lor \neg p (z, y) \bigr ]} \land \color{red}{\neg} \color{green}{p(x,y)} \Bigr ] \Biggr ] \\[1em]
\equiv&\quad \exists x \forall y \Biggl [ \Bigl [\color{blue}{\neg \: \exists z \bigl [ p (y, z) \land p (z, y) \bigr ]} \land \color{green}{p(x,y)} \Bigr ] \; \lor \; \Bigl [ \exists z \bigl [ p (y, z) \lor \neg p (z, y) \bigr ] \land \neg p(x,y) \Bigr ] \Biggr ]
\end{align}$$
Conversion not complete...