Consider the following premises. forall x_8: P_8(x_8) -> Q_8(x_8). forall x_8: P_4(x_8) -> Q_4(x_8). forall x_8: P_4(x_8). forall x_8: P_8(x_8). Can we infer Q_8(a_2) from them? If possible, name the inference rules being used at each step.
 From forall x_8: P_8(x_8) -> Q_8(x_8), forall x_8: P_8(x_8) we can infer forall x_8: Q_8(x_8) via universal modus ponens. From forall x_8: P_4(x_8) -> Q_4(x_8), forall x_8: P_4(x_8) we can infer forall x_8: Q_4(x_8) via universal modus ponens. From forall x_8: Q_4(x_8), forall x_8: Q_8(x_8) we can infer forall x_8: Q_4(x_8) and Q_8(x_8) via universal conjunction. From forall x_8: Q_4(x_8) and Q_8(x_8) we can infer Q_4(a_2) and Q_8(a_2) via universal instantiation. Finally, from Q_4(a_2) and Q_8(a_2) we can infer Q_8(a_2) via simplification. Therefore, the answer is yes.