Consider the following premises. forall z_4: P_2(z_4) -> P(z_4). exists z_4: P_2(z_4) <-> P(z_4). Can we infer forall z_4: ~(P(z_4) <-> P_2(z_4)) from them?
 From exists z_4: P_2(z_4) <-> P(z_4) we can infer exists z_4: P(z_4) -> P_2(z_4). Finally, from exists z_4: P(z_4) -> P_2(z_4), forall z_4: P_2(z_4) -> P(z_4) we can infer exists z_4: P(z_4) <-> P_2(z_4), which contradicts forall z_4: ~(P(z_4) <-> P_2(z_4)). Therefore, the answer is no.