Consider the following premises. forall z_1: T(z_1) -> P_1(z_1). forall z_1: W_2(z_1) <-> T(z_1). forall z_1: P_1(z_1) -> W_2(z_1). Can we infer exists z_1: ~(W_2(z_1) -> P_1(z_1)) from them?
 From forall z_1: W_2(z_1) <-> T(z_1) we can infer forall z_1: W_2(z_1) -> T(z_1). From forall z_1: W_2(z_1) -> T(z_1), forall z_1: T(z_1) -> P_1(z_1) we can infer forall z_1: W_2(z_1) -> P_1(z_1). From forall z_1: P_1(z_1) -> W_2(z_1), forall z_1: W_2(z_1) -> P_1(z_1) we can infer forall z_1: P_1(z_1) <-> W_2(z_1). Finally, from forall z_1: P_1(z_1) <-> W_2(z_1) we can infer forall z_1: W_2(z_1) -> P_1(z_1), which contradicts exists z_1: ~(W_2(z_1) -> P_1(z_1)). Therefore, the answer is no.