Consider the following premises. forall y: S_1(y) or ~P_1(y). forall y: ~S_1(y). forall y: S_1(y) -> P_1(y). Can we infer exists y: P_1(y) from them?
 From forall y: S_1(y) or ~P_1(y) we can infer forall y: P_1(y) -> S_1(y). From forall y: P_1(y) -> S_1(y), forall y: S_1(y) -> P_1(y) we can infer forall y: P_1(y) <-> S_1(y). Finally, from forall y: P_1(y) <-> S_1(y), forall y: ~S_1(y) we can infer forall y: ~P_1(y), which contradicts exists y: P_1(y). Therefore, the answer is no.