$\forall x\exists y \bigl(P(x)\to P(y)\bigr)\to\forall x\exists y\bigl(P(x)\to(y)\bigr)$
Here's what I have so far, but I think it's wrong: $\begin{align*} &\neg\Bigl( \forall x\exists y\bigl(P(x)\to P(y)\bigr) \to \forall x\exists y\bigl( P(x)\to (y)\bigr)\Bigr) &&\text{implication}\\ &\neg\forall x\exists y \bigl( P(x)\to P(y)\bigr)\lor \forall x\exists y \bigl( P(x)\to (y)\bigr) &&\text{implication}\\ &\forall x\exists y\bigl( P(x)\to(y)\bigr) \lor \neg\forall x\exists y\bigl( P(x)\to P(y)\bigr) \equiv \text{TRUE} &&\text{negation} \end{align*}$