Taking the difference of index families of classes I have a question about the conclusion of a proof I am working on. The problem is from Charles Pinter's 'A book on Set Theory'. The problem is as follows:  

Let $\{ A_i \}_{i \in I}$ and $\{ B_j \}_{j \in J}$ be index families of classes. Prove that:
  $$ 
\left( \bigcup_{i \in I} A_i \right) - \left( \bigcup_{j \in J} B_j \right) = \bigcup_{i \in I} \left(\bigcap_{j \in J} \left[ A_i - B_j \right] \right) 
$$

My proof goes as follows:
$$ \begin{align} x \in \left( \bigcup_{i \in I} A_i \right) - \left( \bigcup_{j \in J} B_j \right) & \iff x \in \left( \bigcup_{i \in I} A_i \right) \cap {\left( \bigcup_{j \in J} B_j \right)}^\complement \tag{1}
\\& \iff  x \in \left( \bigcup_{i \in I} A_i \right) \cap \left( \bigcap_{j \in J} {B_j}^\complement \right) \tag{2}
\\ & \iff x \in A_i, \exists i \in I, \land x \in {B_j}^\complement , \forall j \in J \tag{3}
\\ & \iff x \in A_i \cap {B_j}^\complement , \exists i \in I, \forall j \in J \tag{4}
\\&  \iff x \in A_i - B_j , \exists i \in I, \forall j \in J \tag{5}
\\& \iff x \in \bigcup_{i \in I} \left(\bigcap_{j \in J} \left[ A_i - B_j \right] \right) \tag{6} \end{align}$$
$ \qquad \qquad \qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \Box $
But wouldn't it also be correct to conclude $x \in \bigcap_{j \in J} \left(\bigcup_{i \in I} \left[ A_i - B_j \right] \right)$ ? Doesn't the order of operation matter in this case?
 A: Yes, the order of operation does matter in this case. 

As 
  $$x \in A_i - B_j , \exists i \in I, \forall j \in J$$  doesn't imply $$x \in \bigcap_{j \in J} \left(\bigcup_{i \in I} \left[ A_i - B_j \right] \right) $$

You can easily find a counterexample to show this. 

What I think is giving you trouble is the order of quantifiers. 
To show that the order of quantifier does matter, consider the following two statements:-
$$ \forall x \in \mathbb{R} , \exists y \in \mathbb{R}, y^5=x\tag{1}$$
$$ \exists y \in \mathbb{R}, \forall x \in \mathbb{R} , y^5=x\tag{2}$$
Let us consider the translations of both the sentences into words.
$(1)$ "For all $x \in \mathbb{R}$, there exists a $y \in \mathbb{R}$ so that $y^5=x$"
This statement is true, for no matter what $x$ is there exists a number $y=\sqrt[5]{x}$ for which $y^5=x$.
$(2)$ "There exists a $y \in \mathbb{R}$,for all $x \in \mathbb{R}$, so that $y^5=x$"
Another of way of saying this would be that there exists a particular $y$ with the property that $y^5=x$ for every real number $x$.
Clearly, this statement is false. 
Can you take it from here?
