Does the order of establishing domains change the way you read something?

A few weeks ago, just as we had started the semester, I read a question that asked whether these two sentences were the same:

∃ x ∈ A, ∀ y ∈ A, P(x, y) ⇒ Q(x)

∀ y ∈ A, ∃ x ∈ A, P(x, y) ⇒ Q(x)

I remember thinking that these two sentences might be the same in one of those fashions were the order of the factors does not affect the result. Oh boy was I wrong! It is very easy to notice the difference when we put it in words:

For some x that belong to A, all y that belong to A satisfy this...

For all y that belong to A, there is an x that belongs to A such that...

I tried not to dwell too much into the latter part of the equation since it is not relevant to our point. The final point being that it does not only make a difference the way in which the variables and domains are established, but also that it becomes easy to understand once you try it out in English!