\begin{align*}
  \hat{\varphi}_{EUF} \quad := \quad f_x=y \land x = g_x \lor x \neq f_x \land g_x = f_{g_x} \lor y \neq g_x \land x = f_y \land g_y = f_{g_x}
\end{align*}

\begin{align*}
     \varphi_{FC} \quad := \quad & (x=y \rightarrow g_x = g_y)~\land \\
                                 & (x=y \rightarrow f_x = f_y)~\land \\
                                 & (x=g_x \rightarrow f_x = f_{g_x})~\land \\
                                 & (y=g_x \rightarrow f_y = f_{g_x})
\end{align*}
\begin{align*}
     \varphi_{E} \quad := \hat{\varphi}_{EUF} \wedge \varphi_{FC}
\end{align*}