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