Consider the following premises. exists y: Q_1(y) <-> P(y). forall y: Q_1(y) -> P(y). forall y: ~P(y). Can we infer exists y: ~Q_1(y) from them?
 From exists y: Q_1(y) <-> P(y) we can infer exists y: P(y) -> Q_1(y). From exists y: P(y) -> Q_1(y), forall y: Q_1(y) -> P(y) we can infer exists y: P(y) <-> Q_1(y). Finally, from exists y: P(y) <-> Q_1(y), forall y: ~P(y) we can infer exists y: ~Q_1(y). Therefore, the answer is yes.