Consider the following premises. forall x_9: ~Q_9(x_9) or R_2(x_9). forall x_9: ~P_9(x_9) or U(x_9). forall x_9: P_9(x_9). forall x_9: P_7(x_9) <-> U(x_9). forall x_9: ~P_7(x_9). Can we infer forall x_9: R_2(x_9) from them? If possible, name the inference rules being used at each step.
 From forall x_9: P_9(x_9) we can infer forall x_9: P_9(x_9) or Q_9(x_9) via universal addition. From forall x_9: P_7(x_9) <-> U(x_9), forall x_9: ~P_7(x_9) we can infer forall x_9: ~U(x_9) via universal biconditional elimination. From forall x_9: P_9(x_9) or Q_9(x_9), forall x_9: ~P_9(x_9) or U(x_9) we can infer forall x_9: Q_9(x_9) or U(x_9) via universal resolution. From forall x_9: Q_9(x_9) or U(x_9), forall x_9: ~Q_9(x_9) or R_2(x_9) we can infer forall x_9: U(x_9) or R_2(x_9) via universal resolution. Finally, from forall x_9: U(x_9) or R_2(x_9), forall x_9: ~U(x_9) we can infer forall x_9: R_2(x_9) via universal disjunctive syllogism. Therefore, the answer is yes.