\begin{axiom}\label{phase7_dependent_prefix}
    $\emptyset = \emptyset$.
\end{axiom}

\begin{proposition}\label{phase7_dependent_proof}
    For all $x$ we have $x = x$.
\end{proposition}
\begin{proof}
    Fix $x$.
    Show $x = x$.
    \begin{subproof}
        Follows by assumption.
    \end{subproof}
    Follows by assumption.
\end{proof}