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