Consider the following premises. exists x_7: Q_1(x_7) -> Q_7(x_7). forall x_7: P(x_7) -> Q_7(x_7). forall x_7: Q_7(x_7) -> P(x_7). forall x_7: P(x_7). Can we infer exists x_7: Q_7(x_7) from them? If possible, name the inference rules being used at each step.
 From forall x_7: P(x_7) -> Q_7(x_7), forall x_7: Q_7(x_7) -> P(x_7) we can infer forall x_7: P(x_7) <-> Q_7(x_7) via universal biconditional introduction. From forall x_7: P(x_7) <-> Q_7(x_7) we can infer forall x_7: P(x_7) -> Q_7(x_7) via universal biconditional elimination. From forall x_7: P(x_7) we can infer forall x_7: P(x_7) or Q_1(x_7) via universal addition. Finally, from forall x_7: P(x_7) -> Q_7(x_7), exists x_7: Q_1(x_7) -> Q_7(x_7), forall x_7: P(x_7) or Q_1(x_7) we can infer exists x_7: Q_7(x_7) via existential disjunction elimination. Therefore, the answer is yes.