Consider the following premises. forall x_1: Q(x_1). forall x_1: ~Q(x_1). Can we infer forall x_1: T(x_1) from them? If possible, name the inference rules being used at each step.
 From forall x_1: Q(x_1) we can infer forall x_1: Q(x_1) or T(x_1) via universal addition. Finally, from forall x_1: Q(x_1) or T(x_1), forall x_1: ~Q(x_1) we can infer forall x_1: T(x_1) via universal disjunctive syllogism. Therefore, the answer is yes. Notice, however, that the premises were contradictory, so we can infer anything from them.