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