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\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}
\begin{proposition}\label{are_nouns}
Let $x, y$ be foo bars.
Then $x$ and $y$ are foo bars.
\end{proposition}
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