Consider the following premises. forall x_9: P_9(x_9) -> Q_9(x_9). forall x_8: P_9(x_8). forall x_9: Q_9(x_9) -> P_9(x_9). Can we infer Q_9(a_8) from them? If possible, name the inference rules being used at each step.
 From forall x_9: P_9(x_9) -> Q_9(x_9), forall x_9: Q_9(x_9) -> P_9(x_9) we can infer forall x_9: P_9(x_9) <-> Q_9(x_9) via universal biconditional introduction. From forall x_8: P_9(x_8) we can infer P_9(a_8) via universal instantiation. From forall x_9: P_9(x_9) <-> Q_9(x_9) we can infer P_9(a_8) <-> Q_9(a_8) via universal instantiation. From P_9(a_8) <-> Q_9(a_8) we can infer P_9(a_8) -> Q_9(a_8) via biconditional elimination. Finally, from P_9(a_8) -> Q_9(a_8), P_9(a_8) we can infer Q_9(a_8) via modus ponens. Therefore, the answer is yes.