Consider the following premises. forall x: P_4(x) -> W(x). forall x: Q_2(x) -> P_4(x). forall x: W(x) -> Q_2(x). Can we infer exists x: ~(P_4(x) <-> W(x)) from them?
 From forall x: W(x) -> Q_2(x), forall x: Q_2(x) -> P_4(x) we can infer forall x: W(x) -> P_4(x). Finally, from forall x: P_4(x) -> W(x), forall x: W(x) -> P_4(x) we can infer forall x: P_4(x) <-> W(x), which contradicts exists x: ~(P_4(x) <-> W(x)). Therefore, the answer is no.