Nest quantifiers and determining truth value

I must be misunderstanding something...

Here's the issue. I'm going to use A and E as my universal and existential quantifiers because I don't know how to type upside down A's and backwards E's.

Q(x,y) is "x + y = x - y"; domain is all integers

AxEyQ(x,y)

For every integer x there is a integer y such that Q(x,y).

My understanding is...this is true when, for every x there is a y for which Q(x,y) is true. Looking at x + y = x - y...if y=0, then x=x. So this whole thing should be true. Correct?

BUT...my other understand is that, when there is an x such that Q(x,y) is false for every y, then the whole thing is false. Is this correct? So if you let x=0....then y=-y, which is always false. So then the whole thing is false.

It can't be both! What am I not getting right?

Thanks!