a false proof (FO problem)

P(. , .) is a binary relation and f is a unary function.
We want to show that
|= (x=y) -> (P(z,f(x)) <-> P(z, f(y)))

The following is a false proof. But where is the flaw?

Proof. By Deduction Theorem, it suffices to show that
(x=y) |= P(z,f(x)) <-> P(z,f(y))
And according to our convention, A|=phi(x) means A|=forall x phi(x), so
we need to verify that
For any {P, f} model A,
A |= forall x, y (x=y) => A |= forall x,y,z (P(z,f(x)) <-> P(z,f(y))).
But A |= forall x, y (x=y) implies that A has only one element, hence
RHS holds.

.

Loading