Consider the following premises. forall z: R_4(z) or ~P_2(z). forall z: R_4(z) <-> P_2(z). Can we infer exists z: ~(R_4(z) -> P_2(z)) from them?
 From forall z: R_4(z) <-> P_2(z) we can infer forall z: R_4(z) -> P_2(z). From forall z: R_4(z) or ~P_2(z) we can infer forall z: P_2(z) -> R_4(z). From forall z: R_4(z) -> P_2(z), forall z: P_2(z) -> R_4(z) we can infer forall z: R_4(z) <-> P_2(z). Finally, from forall z: R_4(z) <-> P_2(z) we can infer forall z: R_4(z) -> P_2(z), which contradicts exists z: ~(R_4(z) -> P_2(z)). Therefore, the answer is no.