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