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\begin{align*}
\varphi_{FC} \quad := \quad & (x=y \rightarrow g_x = g_y) \wedge \\
& (g_x=y \rightarrow f_{gx} = f_y) \wedge \\
&(g_x=z \rightarrow f_{gx} = f_z) \wedge \\
&(y=z \rightarrow f_y = f_z)
\end{align*}
\begin{align*}
\hat{\varphi}_{EUF} \quad := \quad f_{gx}=f_y \; \lor \; (z=g_y \land z \neq f_z)
\end{align*}
\begin{align*}
\varphi_{E} \quad := \hat{\varphi}_{EUF} \wedge \varphi_{FC}
\end{align*}