Consider the following premises. forall z_1: W(z_1) -> T(z_1). exists z_1: W(z_1) <-> P_2(z_1). forall z_1: T(z_1) -> P_2(z_1). Can we infer forall z_1: ~(P_2(z_1) <-> W(z_1)) from them?
 From exists z_1: W(z_1) <-> P_2(z_1) we can infer exists z_1: P_2(z_1) -> W(z_1). From forall z_1: W(z_1) -> T(z_1), forall z_1: T(z_1) -> P_2(z_1) we can infer forall z_1: W(z_1) -> P_2(z_1). Finally, from exists z_1: P_2(z_1) -> W(z_1), forall z_1: W(z_1) -> P_2(z_1) we can infer exists z_1: P_2(z_1) <-> W(z_1), which contradicts forall z_1: ~(P_2(z_1) <-> W(z_1)). Therefore, the answer is no.