Consider the following premises. forall x_14: P_14(x_14) or Q_14(x_14). ~Q_14(a_12). forall x_14: ~P_14(x_14). Can we infer w_3 from them? If possible, name the inference rules being used at each step.
 From forall x_14: P_14(x_14) or Q_14(x_14), forall x_14: ~P_14(x_14) we can infer forall x_14: Q_14(x_14) via universal disjunctive syllogism. From forall x_14: Q_14(x_14) we can infer Q_14(a_12) via universal instantiation. From Q_14(a_12) we can infer Q_14(a_12) or w_3 via addition. Finally, from Q_14(a_12) or w_3, ~Q_14(a_12) we can infer w_3 via disjunctive syllogism. Therefore, the answer is yes. Notice, however, that the premises were contradictory, so we can infer anything from them.