Consider the following premises. forall x_5: P_5(x_5) <-> Q_2(x_5). forall x_5: ~P_5(x_5). forall x_5: Q_2(x_5). Can we infer forall x_5: Q_1(x_5) from them? If possible, name the inference rules being used at each step.
 From forall x_5: P_5(x_5) <-> Q_2(x_5), forall x_5: ~P_5(x_5) we can infer forall x_5: ~Q_2(x_5) via universal biconditional elimination. From forall x_5: Q_2(x_5) we can infer forall x_5: Q_2(x_5) or Q_1(x_5) via universal addition. Finally, from forall x_5: Q_2(x_5) or Q_1(x_5), forall x_5: ~Q_2(x_5) we can infer forall x_5: Q_1(x_5) via universal disjunctive syllogism. Therefore, the answer is yes. Notice, however, that the premises were contradictory, so we can infer anything from them.