Consider the following premises. forall x_14: R_1(x_14) -> ~Q_9(x_14). forall x_14: P_9(x_14). forall x_14: R_1(x_14) <-> S(x_14). forall x_14: ~P_9(x_14). Can we infer forall x_14: ~S(x_14) from them? If possible, name the inference rules being used at each step.
 From forall x_14: R_1(x_14) <-> S(x_14) we can infer forall x_14: S(x_14) -> R_1(x_14) via universal biconditional elimination. From forall x_14: P_9(x_14) we can infer forall x_14: P_9(x_14) or Q_9(x_14) via universal addition. From forall x_14: P_9(x_14) or Q_9(x_14), forall x_14: ~P_9(x_14) we can infer forall x_14: Q_9(x_14) via universal disjunctive syllogism. From forall x_14: R_1(x_14) -> ~Q_9(x_14), forall x_14: Q_9(x_14) we can infer forall x_14: ~R_1(x_14) via universal modus tollens. Finally, from forall x_14: S(x_14) -> R_1(x_14), forall x_14: ~R_1(x_14) we can infer forall x_14: ~S(x_14) via universal modus tollens. Therefore, the answer is yes. Notice, however, that the premises were contradictory, so we can infer anything from them.