Consider the following premises. forall x_17: ~P_17(x_17). forall x_17: ~Q_17(x_17). forall x_17: P_17(x_17) or Q_17(x_17). u_3 -> ~Q_14(a_1). Can we infer ~u_3 from them? If possible, name the inference rules being used at each step.
 From forall x_17: P_17(x_17) or Q_17(x_17), forall x_17: ~P_17(x_17) we can infer forall x_17: Q_17(x_17) via universal disjunctive syllogism. From forall x_17: Q_17(x_17) we can infer forall x_17: Q_17(x_17) or Q_14(x_17) via universal addition. From forall x_17: Q_17(x_17) or Q_14(x_17), forall x_17: ~Q_17(x_17) we can infer forall x_17: Q_14(x_17) via universal disjunctive syllogism. From forall x_17: Q_14(x_17) we can infer Q_14(a_1) via universal instantiation. Finally, from u_3 -> ~Q_14(a_1), Q_14(a_1) we can infer ~u_3 via modus tollens. Therefore, the answer is yes. Notice, however, that the premises were contradictory, so we can infer anything from them.