I doubt it is possible to find the same domain for all three quantifiers. How can one avoid instantiating z, in particular, to x?

In fact, I used my favorite proof assistant program to prove the negation of this statement. The actual code (the proof is omitted) is the following.

Code:Parameter T : Type. Theorem t : ~ (exists x : T, exists y : T, (x <> y /\ forall z : T, (z <> x /\ z <> y))). Proof. ... Qed.