Suppose $A$ is an arbitrary nonempty set and assume $P, Q$ and $R$ are predicates taking in elements of $\mathbb{Z}$ and $A$ as inputs. Also, suppose that the following predicate logic statement is always true.
$$\exists x \in \mathbb{Z} \ \forall y \in \mathbb{N} \ [[(\forall z \in \mathbb{Z}, P(x,y,z)) \land (\exists w \in \mathbb{N}, Q(x,y,w))] \rightarrow\exists a \in A \ \forall b \in A \ , R(x,y,a,b)]$$
Does the value of $a$ depend on $z$? How about $w$?
Not completely sure about this.
I'm guessing that $a$ doesn't depend on $z$, because $P(x,y,z)$ is assumed true for all integers $z$. Hence we can choose a 'uniform' $a$ independent of $z$. However, $a$ may depend on $w$, since $w$ is 'fixed' in a sense.
$a$ does not depend on either $z$ or $w$, since it does not occur in the scope of the quantifiers of either $w$ or $z$.
Another way of seeing this is that we can pull out the $\exists a \in A$ using the Prenex laws. That is, the statement is equivalent to:
$$\exists x \in \mathbb{Z} \ \forall y \in \mathbb{N} \ \exists a \in A \ [[(\forall z \in \mathbb{Z}, P(x,y,z)) \land (\exists w \in \mathbb{N}, Q(x,y,w))] \rightarrow\ \forall b \in A \ , R(x,y,a,b)]$$
and now it is real clear that $a$ does not depend on either $w$ or $z$