\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*}