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This sequent is not provable.
Model $\mathcal{M}$:\\
\begin{minipage}{0.2\textwidth}
\begin{align*}
\mathcal{A} =& \; \{ a,b \}\\
P^\mathcal{M} =& \; \{ a \}\\
Q^\mathcal{M} =& \; \{ a \} \\
y \leftarrow \; b \\
\end{align*}
\end{minipage}\\
$\mathcal{M} \models \exists x \; (P(x) \imp Q(y)), \quad \exists x \; P(x)$\\
$\mathcal{M} \nmodels Q(y)$