About the notation of "there exist" and "For all" I have a question like this

Let $P(x,y,z)$ denote $xy=z$, $E(x,y)$ denote $x=y$, $G(x,y)$ denote $x>y$. Transcribe the following into logical notation. Assume that the universe of discourse is the set of integers.
(1) If $y=1$, then $xy=x$

The solution given is: $\forall y [E(y,1)\rightarrow \forall xP(x,y,x)]  $
Why is it not $\exists y [E(y,1)\rightarrow \forall xP(x,y,x)]  $ ?
I though the $y$ only have the value of $1$, why using $\forall$??

The original solution is correct for the reasons explained by Brian Scott. It is, as he says, the correct interpretation of his sentence 2.
And sentence 2 is exactly like the original, except that sentence two makes the domain of integers explicit.
