Consider the following premises. exists x_3: ~P_3(x_3) or R_1(x_3). forall x_3: ~Q_4(x_3). forall x_3: P_3(x_3) or Q_4(x_3). Can we infer exists x_3: R_1(x_3) from them? If possible, name the inference rules being used at each step.
 From forall x_3: P_3(x_3) or Q_4(x_3), exists x_3: ~P_3(x_3) or R_1(x_3) we can infer exists x_3: Q_4(x_3) or R_1(x_3) via existential resolution. Finally, from exists x_3: Q_4(x_3) or R_1(x_3), forall x_3: ~Q_4(x_3) we can infer exists x_3: R_1(x_3) via existential disjunctive syllogism. Therefore, the answer is yes.