Consider the following premises. forall x_9: ~P_9(x_9) or R_9(x_9). forall x_9: P_9(x_9) or Q_9(x_9). forall x_9: R_7(x_9) -> S_2(x_9). forall x_9: R_9(x_9) -> S_2(x_9). forall x_9: ~Q_9(x_9) or R_7(x_9). Can we infer exists z: S_2(z) from them? If possible, name the inference rules being used at each step.
 From forall x_9: P_9(x_9) or Q_9(x_9), forall x_9: ~P_9(x_9) or R_9(x_9) we can infer forall x_9: Q_9(x_9) or R_9(x_9) via universal resolution. From forall x_9: Q_9(x_9) or R_9(x_9), forall x_9: ~Q_9(x_9) or R_7(x_9) we can infer forall x_9: R_9(x_9) or R_7(x_9) via universal resolution. From forall x_9: R_9(x_9) -> S_2(x_9), forall x_9: R_7(x_9) -> S_2(x_9), forall x_9: R_9(x_9) or R_7(x_9) we can infer forall x_9: S_2(x_9) via universal disjunction elimination. From forall x_9: S_2(x_9) we can infer S_2(i) via universal instantiation. Finally, from S_2(i) we can infer exists z: S_2(z) via existential generalization. Therefore, the answer is yes.