Consider the following premises. forall x: P(x) -> S_1(x). exists x: ~T_4(x). forall x: P(x) <-> T_4(x). forall x: S_1(x) -> P(x). Can we infer exists x: ~S_1(x) from them?
 From forall x: P(x) <-> T_4(x), exists x: ~T_4(x) we can infer exists x: ~P(x). From forall x: P(x) -> S_1(x), forall x: S_1(x) -> P(x) we can infer forall x: P(x) <-> S_1(x). Finally, from forall x: P(x) <-> S_1(x), exists x: ~P(x) we can infer exists x: ~S_1(x). Therefore, the answer is yes.