\item Consider the following natural deduction proof for the sequent $$\exists x \; \lnot P(x) \quad \ent \quad \lnot \forall x \; P(x).$$
Is the proof correct? If not, explain the error in the proof and either show how to correctly prove the sequent, or give a counterexample that proves the sequent invalid.

\setlength\subproofhorizspace{1em}
\begin{logicproof}{1}
  \exists x \; \lnot P(x)           & prem.\\
  \begin{subproof}
    \forall x \; P(x)		            & ass.\\
    P(x_0)                          & $\forall \mathrm{e}$ 2\\
    \exists x \; P(x)               & $\exists \mathrm{i}$ 3\\
    \bot                            & $\lnot \mathrm{e}$ 1,4
  \end{subproof}
  \lnot \forall x \; P(x)           & $\lnot \mathrm{e}$ 2-5
\end{logicproof}