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\} \\
  R^\mathcal{M} =& \; \{ \}
  \end{align*}
\end{minipage}\\

$\mathcal{M} \models \exists x (Q(x) \imp R(x))$\\
$\mathcal{M} \models \exists x (P(x) \land Q(x))$\\
$\mathcal{M} \nmodels \exists x (P(x) \land R(x))$