Is this equivalence true?
$(\forall x (P(x)) \wedge (\exists y Q(y)) \equiv \forall x \exists y(P(x) \wedge \exists x Q(y))$
Here is what I did so far.
If the LHS is true, then there exists a x such that P(x) is true and a y such that Q(y) is true.
If the RHS is false, then there exists a x such that P(x) is false and a y such that Q(y) is false.
Thus both statements are equivalent.