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$\forall x\exists y \bigl(P(x)\to P(y)\bigr)\to\forall x\exists y\bigl(P(x)\to(y)\bigr)$

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

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    Note that $\neg(P\to Q)\equiv P\land \neg Q$. Instead, you have $\neg(P\to Q)\equiv \neg P\lor Q$, which is incorrect. If your second line was meant to be $\neg(\neg P\lor Q)$, then you are missing a negation, and the negation of *that* would be of the form $P\land \neg Q$.2012-01-17

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