Consider the following premises. forall x_14: Q_14(x_14) -> P_14(x_14). exists x_14: ~Q_14(x_14). forall x_14: P_14(x_14). forall x_14: P_14(x_14) <-> Q_14(x_14). Can we infer exists x_14: Q_1(x_14) from them? If possible, name the inference rules being used at each step.
 From forall x_14: P_14(x_14) <-> Q_14(x_14) we can infer forall x_14: P_14(x_14) -> Q_14(x_14) via universal biconditional elimination. From forall x_14: Q_14(x_14) -> P_14(x_14), forall x_14: P_14(x_14) -> Q_14(x_14) we can infer forall x_14: Q_14(x_14) <-> P_14(x_14) via universal biconditional introduction. From forall x_14: Q_14(x_14) <-> P_14(x_14), exists x_14: ~Q_14(x_14) we can infer exists x_14: ~P_14(x_14) via existential biconditional elimination. From forall x_14: P_14(x_14) we can infer forall x_14: P_14(x_14) or Q_1(x_14) via universal addition. Finally, from forall x_14: P_14(x_14) or Q_1(x_14), exists x_14: ~P_14(x_14) we can infer exists x_14: Q_1(x_14) via existential disjunctive syllogism. Therefore, the answer is yes. Notice, however, that the premises were contradictory, so we can infer anything from them.