Consider the following premises. exists y: Q(y) -> Q_4(y). forall y: P(y) <-> Q_4(y). forall y: Q_4(y) -> Q(y). forall y: ~P(y). Can we infer forall y: Q(y) from them? If possible, name the inference rules being used at each step.
 From forall y: P(y) <-> Q_4(y), forall y: ~P(y) we can infer forall y: ~Q_4(y) via universal biconditional elimination. 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) via existential biconditional introduction. Finally, from exists y: Q_4(y) <-> Q(y), forall y: ~Q_4(y) we can infer exists y: ~Q(y) via existential biconditional elimination, which contradicts forall y: Q(y). Therefore, the answer is no.