Consider the following premises. forall x_12: R(x_12) -> P(x_12). forall x_12: P_12(x_12) or Q_12(x_12). forall x_12: Q_12(x_12) -> R(x_12). forall x_12: P(x_12) -> Q_1(x_12). forall x_12: P_12(x_12) -> R(x_12). Can we infer Q_1(a) from them? If possible, name the inference rules being used at each step.
 From forall x_12: P_12(x_12) -> R(x_12), forall x_12: Q_12(x_12) -> R(x_12), forall x_12: P_12(x_12) or Q_12(x_12) we can infer forall x_12: R(x_12) via universal disjunction elimination. From forall x_12: R(x_12) -> P(x_12), forall x_12: R(x_12) we can infer forall x_12: P(x_12) via universal modus ponens. From forall x_12: P(x_12) -> Q_1(x_12), forall x_12: P(x_12) we can infer forall x_12: Q_1(x_12) via universal modus ponens. Finally, from forall x_12: Q_1(x_12) we can infer Q_1(a) via universal instantiation. Therefore, the answer is yes.