I'm trying to understand whether a certain free variable is necessary in versions of the Separation and Replacement axioms, as Kunen and Jech seem to give different versions. For example, Kunen's version of Separation is
$\forall p_1 \cdots \forall p_k \forall A \exists B \forall x (x \in B \leftrightarrow (x \in A \wedge \varphi(x,A,p_1,\dots,p_k)))$,
while Jech's version seems to not allow $A$ to be free in $\varphi$. There is some great information about how one can drop the parameters $p_1,\dots,p_k$ in answers to this question, but they do not seem to answer the question about $A$. Informally, we do often write sets like $\{x \in A : x^2 \in A\}$, which seems to require allowing reference to $A$ in our formula.
My question is whether these two versions are equivalent, relative to the other axioms. And I also have the same question for the Replacement Axiom, as Kunen also seems to allow $A$ to be a free variable in the formula defining the class function with domain $A$.
