\begin{definition}\label{bar}
    $x$ is a bar iff $x = x$.
\end{definition}

\begin{definition}\label{foo}
    $x$ is foo iff $x = x$.
\end{definition}

\begin{definition}\label{baz}
    $x$ is baz iff $x = x$.
\end{definition}

\begin{proposition}\label{nouns}
    Let $x, y$ be bars.
    Then $x = x$.
\end{proposition}

\begin{proposition}\label{adj_nouns}
    Let $x, y$ be foo bars.
    Then $x = x$.
\end{proposition}


\begin{proposition}\label{nouns_suchthat}
    Let $x, y$ be bars such that $x$ is foo and $y$ is baz.
    Then $x = x$.
\end{proposition}


\begin{proposition}\label{noun_verb}
    $x = y$ iff $x$ is a bar equal to $y$.
\end{proposition}


\begin{proposition}\label{adjs}
    $x$ is foo and baz.
\end{proposition}
