Consider the following premises. forall x_3: P_3(x_3). exists x_3: ~P(x_3) or R(x_3). forall x_3: P_3(x_3) -> P(x_3). Can we infer exists x_3: Q_1(x_3) or R(x_3) from them? If possible, name the inference rules being used at each step.
 From forall x_3: P_3(x_3) -> P(x_3), forall x_3: P_3(x_3) we can infer forall x_3: P(x_3) via universal modus ponens. From forall x_3: P(x_3) we can infer forall x_3: P(x_3) or Q_1(x_3) via universal addition. Finally, from forall x_3: P(x_3) or Q_1(x_3), exists x_3: ~P(x_3) or R(x_3) we can infer exists x_3: Q_1(x_3) or R(x_3) via existential resolution. Therefore, the answer is yes.