Consider the following premises. forall x_13: Q_13(x_13). forall x_13: P_13(x_13) -> P_1(x_13). forall x_13: P_4(x_13) -> R_2(x_13). forall x_13: Q_4(x_13) -> R_2(x_13). forall x_13: P_4(x_13) or Q_4(x_13). forall x_13: P_13(x_13). Can we infer forall x_13: P_1(x_13) and R_2(x_13) from them? If possible, name the inference rules being used at each step.
 From forall x_13: P_13(x_13), forall x_13: Q_13(x_13) we can infer forall x_13: P_13(x_13) and Q_13(x_13) via universal conjunction. From forall x_13: P_13(x_13) and Q_13(x_13) we can infer forall x_13: P_13(x_13) via universal simplification. From forall x_13: P_13(x_13) -> P_1(x_13), forall x_13: P_13(x_13) we can infer forall x_13: P_1(x_13) via universal modus ponens. From forall x_13: P_4(x_13) -> R_2(x_13), forall x_13: Q_4(x_13) -> R_2(x_13), forall x_13: P_4(x_13) or Q_4(x_13) we can infer forall x_13: R_2(x_13) via universal disjunction elimination. Finally, from forall x_13: P_1(x_13), forall x_13: R_2(x_13) we can infer forall x_13: P_1(x_13) and R_2(x_13) via universal conjunction. Therefore, the answer is yes.