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