Consider the following premises. forall x_9: Q_9(x_9) -> S_2(x_9). forall x_9: ~P_3(x_9). forall x_9: P_9(x_9) -> S_2(x_9). forall x_9: P_3(x_9) or S(x_9). forall x_9: P_9(x_9) or Q_9(x_9). Can we infer exists x_9: ~S(x_9) from them? If possible, name the inference rules being used at each step.
 From forall x_9: P_9(x_9) -> S_2(x_9), forall x_9: Q_9(x_9) -> S_2(x_9), forall x_9: P_9(x_9) or Q_9(x_9) we can infer forall x_9: S_2(x_9) via universal disjunction elimination. From forall x_9: P_3(x_9) or S(x_9), forall x_9: ~P_3(x_9) we can infer forall x_9: S(x_9) via universal disjunctive syllogism. From forall x_9: S_2(x_9), forall x_9: S(x_9) we can infer forall x_9: S_2(x_9) and S(x_9) via universal conjunction. Finally, from forall x_9: S_2(x_9) and S(x_9) we can infer forall x_9: S_2(x_9) via universal simplification, which contradicts exists x_9: ~S(x_9). Therefore, the answer is no.