Valid logical statements

Question

I was wondering if the following is a valid statement:

$\forall x \exists y\ (y\neq x)\ P(x,y)$, where P(x) is 'x is the parent of y' and the domain of x and y is the set of all animals.

I am specifically referring to the part '$(y\neq x)\ P(x,y)$'.

Answer 1

The expression isn't well-formed, though the intent is reasonably clear. The proper version is $\forall x\exists y(\lnot(x=y)\land P(x,y))$. And this sentence happens to be false.