It's not true.
When you say "for all y there exists x such that.." you're saying that x depends on y. In other words, you're saying that you choose any value for y, and then you choose an adequate value for x.
On the other hand, claiming that "there exists x such that, for all y's, ...", you're talking about the existence of a universal value for x, which is "good" for every possible value of y.
Think of this as a role play:
means that your friend picks a value for y, and you answer choosing a value for x accordingly.
On the other hand,
means that you can pick a value for x that will make you "win" no matter which y-value your friend chooses.
For the counter example: let P(x,y) be "x is the double of y".
It is true that
for every number y your friends chooses, you pick its double: your friend chooses y=4 and you pick x=8, your friend chooses y=-5 and you pick -10, and so on.
But it is not true that you can pick a unique value for x, and that it will be the double of whatever number your friend will pick: if you choose x=6 you will win only if your friend chooses y=3, in all the other cases you will lose.