I am trying to show that the following are not logically equivalent (according to a practice question)
$\exists x \forall y (P(y) \implies Q(x))$ and $\forall y P(y) \implies \exists x Q(x)$
In the first case I am trying to find some kind of statement where $x,y$ are integers (something like $P(x)$ is the is even predicate and $Q(x)$ is odd predicate, or maybe that $Q(x)$ implies $x$ divides $y$).
I am imagining I need a scenario where one statement is True implies False, and the other is True, for the same values of $x,y$.
If $\forall y P(y)$ is false, then both implications will be true, so suppose that $\forall y$ P(y) is true. I'm not sure how to proceed from here. Hints/Clarifications appreciated.