Consider the following premises. forall z: R_4(z) <-> P_3(z). forall z: ~R_4(z). forall z: P_3(z) -> R_4(z). Can we infer exists z: P_3(z) from them?
 From forall z: R_4(z) <-> P_3(z) we can infer forall z: R_4(z) -> P_3(z). From forall z: R_4(z) -> P_3(z), forall z: P_3(z) -> R_4(z) we can infer forall z: R_4(z) <-> P_3(z). Finally, from forall z: R_4(z) <-> P_3(z), forall z: ~R_4(z) we can infer forall z: ~P_3(z), which contradicts exists z: P_3(z). Therefore, the answer is no.