Consider the following premises. R(a) -> P_1(a). forall z: P_1(z) <-> R(z). Can we infer R(a) <-> P_1(a) from them?
 From forall z: P_1(z) <-> R(z) we can infer P_1(a) <-> R(a). From P_1(a) <-> R(a) we can infer P_1(a) -> R(a). Finally, from R(a) -> P_1(a), P_1(a) -> R(a) we can infer R(a) <-> P_1(a). Therefore, the answer is yes.