What I did was to first assume
$$ \forall x (P(x) \to \exists y Q(x, y))$$ Assume a free variable $x_0$
then doing a universal quantifier elimination
$$P(x_0) \to \exists y Q(x_0, y)$$
Now assuming $P(x_0)$
Using implication elimination
$$ \exists y Q(x_0, y)$$
This is all I managed to do because I was unable to remove the final line from the scope of $x_0$ assumption since there is no quantifier for x. How should I proceed?