Consider the following premises. forall x_7: P_3(x_7) -> P(x_7). exists x_7: ~P_7(x_7) or S(x_7). forall x_7: ~P(x_7). forall x_7: P_7(x_7) or P_3(x_7). Can we infer exists x_7: S(x_7) from them? If possible, name the inference rules being used at each step.
 From forall x_7: P_7(x_7) or P_3(x_7), exists x_7: ~P_7(x_7) or S(x_7) we can infer exists x_7: P_3(x_7) or S(x_7) via existential resolution. From forall x_7: P_3(x_7) -> P(x_7), forall x_7: ~P(x_7) we can infer forall x_7: ~P_3(x_7) via universal modus tollens. Finally, from exists x_7: P_3(x_7) or S(x_7), forall x_7: ~P_3(x_7) we can infer exists x_7: S(x_7) via existential disjunctive syllogism. Therefore, the answer is yes.