Proper use of Universal Instantiation

Is this a proper use of universal instantiation? If not, how could this proof be corrected?

Suppose:

$\displaystyle \forall x P(x) \rightarrow \exists x Q(x)$

Prove:

$\displaystyle \exists x (P(x) \rightarrow Q(x))$

Proof:

$\displaystyle y$ is arbitray, so $\displaystyle P(y)$

$\displaystyle P(y)$, so choose a $\displaystyle z_{0}$ such that $\displaystyle Q(z_{0})$

By universal instantiation let $\displaystyle y = z_{0}$.

Therefore $\displaystyle P(z_{0}) \rightarrow Q(z_{0})$.

$\displaystyle \square$