Consider the following premises. exists z: S_2(z) <-> S(z). forall z: S_2(z) -> P_3(z). forall z: P_3(z) -> S_2(z). Can we infer exists z: S(z) -> P_3(z) from them?
 From exists z: S_2(z) <-> S(z) we can infer exists z: S(z) -> S_2(z). From forall z: S_2(z) -> P_3(z), forall z: P_3(z) -> S_2(z) we can infer forall z: S_2(z) <-> P_3(z). From forall z: S_2(z) <-> P_3(z) we can infer forall z: S_2(z) -> P_3(z). Finally, from exists z: S(z) -> S_2(z), forall z: S_2(z) -> P_3(z) we can infer exists z: S(z) -> P_3(z). Therefore, the answer is yes.