The schema of separation states: for each set $A$ and each set-theoretic formula $\Phi(x)$ with the only free variable $x$ there exists a set $\{a\in A\;|\;\models\Phi(a)\}$ of elements of $A$ satisfying the property $\Phi.$
Let $A$ be a set. Suppose we want to construct $\bigcap A.$ How do they act sometimes? First, using the axiom of union, one obtains $\bigcup A.$ Than considers $\Phi\eqcirc\forall b\in A:x\in a$ and uses separation: $\bigcap A=\{a\in\bigcup A\;|\;\forall b\in A:a\in b\}.$
My question is about legality of such a reasoning: here $\Phi$ contains both $x$ and $A$ as free variables so it seems invalid. How should this difficulty be avoided?
