Consider the following premises. exists y_2: Q(y_2) -> Q_4(y_2). forall y_2: P(y_2) -> Q(y_2). forall y_2: Q_4(y_2) -> Q(y_2). forall y_2: P(y_2). Can we infer exists y_2: Q(y_2) from them? If possible, name the inference rules being used at each step.
 From forall y_2: Q_4(y_2) -> Q(y_2), exists y_2: Q(y_2) -> Q_4(y_2) we can infer exists y_2: Q_4(y_2) <-> Q(y_2) via existential biconditional introduction. From forall y_2: P(y_2) we can infer forall y_2: P(y_2) or Q_4(y_2) via universal addition. From exists y_2: Q_4(y_2) <-> Q(y_2) we can infer exists y_2: Q_4(y_2) -> Q(y_2) via existential biconditional elimination. Finally, from forall y_2: P(y_2) -> Q(y_2), exists y_2: Q_4(y_2) -> Q(y_2), forall y_2: P(y_2) or Q_4(y_2) we can infer exists y_2: Q(y_2) via existential disjunction elimination. Therefore, the answer is yes.