When is interchange of quantifiers allowed? Ex: $\forall w \in \bigcup A_n$ There is a myriad of question for the interchange of different quantifiers, mainly between $\exists$ and $\forall$.
I'm interested in knowing when both can be interchanged.
The motivation came from this:
$\forall w \in \bigcup_n A_n \Leftrightarrow\forall_{n,w} w \in A_n$, when $w \in \bigcup_n A_n \Leftrightarrow\exists_{n} w \in A_n$.
Any help would be appreciated.
 A: Example 
Suppose that for every man, the there exists a women that is his mother. Symbolically, we can state this relationship as follows:
$$\forall x:[x \in Men \implies \exists y: [y\in Women \land Mother(y,x)]]$$
Or equivalently:
$$\forall x\in Men: \exists y\in Women: Mother(y,x)$$
It should be obvious that we cannot infer from this statement that there exists a woman that is the mother all men:
$$\exists y: [y\in Women \land \forall x:[x \in Men \implies Mother(y,x)]]$$
Or equivalently:
$$\exists y\in Women: \forall x\in Men: Mother(y,x)$$

As a general rule, in mathematical proofs with all quantifiers restricted to various sets like this: 

When discharging a premise (or assumption), the conclusion should not refer to any free variables that were introduced after that premise. And any free variables that were introduced in that premise should be universally generalized in the conclusion. 

Follow this rule and you shouldn't have to worry about rules for interchanging quantifiers.
