Consider the following premises. forall y: T_1(y) -> W_4(y). forall y: P_2(y) -> T_1(y). forall y: P_2(y) <-> W_4(y). Can we infer exists y: ~(P_2(y) <-> W_4(y)) from them?
 From forall y: P_2(y) <-> W_4(y) we can infer forall y: W_4(y) -> P_2(y). From forall y: P_2(y) -> T_1(y), forall y: T_1(y) -> W_4(y) we can infer forall y: P_2(y) -> W_4(y). Finally, from forall y: P_2(y) -> W_4(y), forall y: W_4(y) -> P_2(y) we can infer forall y: P_2(y) <-> W_4(y), which contradicts exists y: ~(P_2(y) <-> W_4(y)). Therefore, the answer is no.