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+\begin{proposition}\label{trivial}
+    $x = x$.
+\end{proposition}
+
+\begin{proposition}\label{irrelevant}
+    $z = z$.
+\end{proposition}
+
+\begin{proposition}\label{alsotrivial}
+    $y = y$.
+\end{proposition}
+\begin{proof}
+    \begin{align*}
+        y
+            &= y
+                \\
+            &= y
+                \explanation{by \cref{trivial}}
+    \end{align*}
+\end{proof}
+
+\begin{proposition}\label{trivial_biconditionals}
+    $y = y$.
+\end{proposition}
+\begin{proof}
+    \begin{align*}
+        y = y
+            &\iff \top
+                \\
+            &\iff y = y
+                \explanation{by \cref{trivial}}
+    \end{align*}
+\end{proof}