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I have working on this formula $(\forall x \exists y (P(x) \supset Q(y))) \supset (\exists y \forall x (P (x) \supset Q(y)))$ to either prove or disprove it.

First, I tempted to disprove it, but I changed my mind.

I wrote down "for all x that there exists some y satisfies corresponding condition", and "there exists some y that for all x satisfies corresponding condition." I think these statements refer to the same idea.

Any suggestions?

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    If you replace the specific clause $P(x)\implies Q(y)$ with an arbitrary formula $\Phi(x,y)$, this formula isn't generally true. The left-hand side means "for every specific $x$, we can find *some* $y$ that makes $\Phi(x,y)$ true". The right-hand side means "there is a particular $y$ that makes $\Phi(x,y)$ true *regardless* of $x$". This is a much stronger statement in general. In other words, the two statements don't refer to the same idea in general.2011-12-08
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    Take a look at the answers for [Confused between Nested Quantifiers](http://math.stackexchange.com/questions/64500/confused-between-nested-quantifiers) It will help you.2011-12-08
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    I have looked Confused between Nested Quantifiers, but there is only one predicate. I think this is different.2011-12-08
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    There are cases where P(x) and Q(y) have different truth values.2011-12-08

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