Consider the following premises. forall y: P_4(y) <-> Q(y). forall y: P_4(y) -> S_3(y). forall y: ~Q(y). exists y: S_3(y) -> P_4(y). Can we infer exists y: ~S_3(y) from them?
 From forall y: P_4(y) <-> Q(y), forall y: ~Q(y) we can infer forall y: ~P_4(y). From exists y: S_3(y) -> P_4(y), forall y: P_4(y) -> S_3(y) we can infer exists y: S_3(y) <-> P_4(y). Finally, from exists y: S_3(y) <-> P_4(y), forall y: ~P_4(y) we can infer exists y: ~S_3(y). Therefore, the answer is yes.