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