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