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\begin{align*} \hat{\varphi}_{EUF} \quad := \quad f_{ab} = x \land f_{xy} \neq g_a \lor f_{mn} = b \lor f_{g_{a}y} \neq a\end{align*}
\begin{align*} \varphi_{FC} \quad := \quad & (a=m \land b=n \rightarrow f_{ab} = f_{mn}) \land \\ & (a=x \land b=y \rightarrow f_{ab} = f_{xy}) \land \\ & (a=g_a\land b=y \rightarrow f_{ab} = f_{g_{a}y}) \land \\ & (m=g_a\land y=n \rightarrow f_{mn} = f_{g_{a}y}) \land \\ & (m=x \land y=n \rightarrow f_{xy} = f_{mn}) \land \\ & (x=g_a\land y=y \rightarrow f_{xy} = f_{g_{a}y})\end{align*}\begin{align*} \varphi_{E} \quad := \hat{\varphi}_{EUF} \wedge \varphi_{FC}\end{align*}
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