Consider the following premises. forall y: P(y) <-> Q_4(y). forall y: ~P(y). forall y: Q_4(y) -> Q(y). exists y: Q(y) -> Q_4(y). Can we infer exists y: ~Q(y) from them?
 From forall y: P(y) <-> Q_4(y), forall y: ~P(y) we can infer forall y: ~Q_4(y). From forall y: Q_4(y) -> Q(y), exists y: Q(y) -> Q_4(y) we can infer exists y: Q_4(y) <-> Q(y). Finally, from exists y: Q_4(y) <-> Q(y), forall y: ~Q_4(y) we can infer exists y: ~Q(y). Therefore, the answer is yes.