Consider the following premises. forall y: S(y) -> P_3(y). forall y: P_1(y) -> S(y). forall y: S(y) -> P_1(y). exists y: ~P_3(y). Can we infer forall y: P_1(y) from them?
 From forall y: P_1(y) -> S(y), forall y: S(y) -> P_1(y) we can infer forall y: P_1(y) <-> S(y). From forall y: S(y) -> P_3(y), exists y: ~P_3(y) we can infer exists y: ~S(y). Finally, from forall y: P_1(y) <-> S(y), exists y: ~S(y) we can infer exists y: ~P_1(y), which contradicts forall y: P_1(y). Therefore, the answer is no.