\begin{proposition}\label{formula_test_forall}
    $\forall x, y. x = x\land y = y$.
\end{proposition}

\begin{proposition}\label{formula_test_exists}
    $\exists x, y. x = y$.
\end{proposition}

\begin{proposition}\label{formula_test_not_exists}
    $\exists x, y. x = y$ if and only if not $\exists x. x \neq x$.
\end{proposition}