Verified answer

Yes, there is. Let's take an easy example:

Let A(x,y) be the statement

x + y = 0

Then ∀x ∃y: A(x,y) translates

"For every x, there is some y such that x + y = 0"

And of course, that's true; for any x, we just let y = -x.

But the statement ∃y ∀x: A(x,y) translates

"There is some y such that for EVERY x, x + y = 0"

and of course, that's not true. There is no number that can be added to ANY number to get 0 as a result.

I hope this helps clear up the distinction.

The qualifications apply left to right, so

∀x ∃y: A(x,y) means that for EVERY x, the statement ∃y: A(x,y) is true, but

∃y ∀x: A(x,y) means that for some y, the statement ∀x: A(x,y) is true.