Consider the following premises. forall x_9: S(x_9) -> Q_2(x_9). forall x_9: P_9(x_9) <-> Q_9(x_9). exists x_9: Q_2(x_9) -> P_9(x_9). forall x_9: ~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) <-> Q_9(x_9), forall x_9: ~Q_9(x_9) we can infer forall x_9: ~P_9(x_9) via universal biconditional elimination. From forall x_9: S(x_9) -> Q_2(x_9), exists x_9: Q_2(x_9) -> P_9(x_9) we can infer exists x_9: S(x_9) -> P_9(x_9) via existential transitivity. Finally, from exists x_9: S(x_9) -> P_9(x_9), forall x_9: ~P_9(x_9) we can infer exists x_9: ~S(x_9) via existential modus tollens. Therefore, the answer is yes.