Generalized Distributive Law

This is problem 3.11, p.29, Introduction to Set Theory, Hrbacek and Jech.

$ $ \big( \bigcap_{a \in A} F_a \big) \cup \big( \bigcap_{b \in B} G_b \big) = \bigcap_{(a,b) \in A \times B} (F_a \cup G_b). $ $

It is my attempt.

$ \begin{equation} x \in (LHS) \ \Leftrightarrow x \in \bigcap_{a \in A} F_a ~~ or ~~ x \in \bigcap_{b \in B} G_b \ \Leftrightarrow (x \in F_a ~~for ~~all~~a\in A)~~or~~ (x\in G_b~~ for~~all~~b\in B). \end{equation} $

And,

$ \begin{equation} x\in(RHS) \ \Leftrightarrow x \in F_a \cup G_b~~for ~~all~~(a,b) \in A \times B \ \Leftrightarrow (x\in F_a~~or ~~ x \in G_b) ~~for ~~all~~(a,b) \in A \times B. \end{equation} $

I want to change the position of phrases in each last sentences, but I’m not sure doing it preserves if and only if condition.

Please help my problem.