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