Consider the following premises. forall x_11: ~Q_11(x_11). forall x_11: Q(x_11) -> Q_11(x_11). forall x_11: Q(x_11) and Q_5(x_11). forall x_11: Q_11(x_11) -> Q(x_11). Can we infer forall x_11: Q_1(x_11) from them? If possible, name the inference rules being used at each step.
 From forall x_11: Q(x_11) -> Q_11(x_11), forall x_11: Q_11(x_11) -> Q(x_11) we can infer forall x_11: Q(x_11) <-> Q_11(x_11) via universal biconditional introduction. From forall x_11: Q(x_11) <-> Q_11(x_11), forall x_11: ~Q_11(x_11) we can infer forall x_11: ~Q(x_11) via universal biconditional elimination. From forall x_11: Q(x_11) and Q_5(x_11) we can infer forall x_11: Q(x_11) via universal simplification. From forall x_11: Q(x_11) we can infer forall x_11: Q(x_11) or Q_1(x_11) via universal addition. Finally, from forall x_11: Q(x_11) or Q_1(x_11), forall x_11: ~Q(x_11) we can infer forall x_11: Q_1(x_11) via universal disjunctive syllogism. Therefore, the answer is yes. Notice, however, that the premises were contradictory, so we can infer anything from them.