Consider the following premises. forall z: P_1(z) <-> S(z). exists z: T_2(z) -> P_1(z). forall z: P(z) -> T_2(z). Can we infer exists z: P(z) -> S(z) from them?
 From forall z: P(z) -> T_2(z), exists z: T_2(z) -> P_1(z) we can infer exists z: P(z) -> P_1(z). From forall z: P_1(z) <-> S(z) we can infer forall z: P_1(z) -> S(z). Finally, from exists z: P(z) -> P_1(z), forall z: P_1(z) -> S(z) we can infer exists z: P(z) -> S(z). Therefore, the answer is yes.