Consider the following premises. forall z: W(z) -> Q_1(z). forall z: W(z). forall z: Q_1(z) -> W(z). exists z: ~Q_1(z). Can we infer exists z: T_4(z) from them? If possible, name the inference rules being used at each step.
 From forall z: Q_1(z) -> W(z), forall z: W(z) -> Q_1(z) we can infer forall z: Q_1(z) <-> W(z) via universal biconditional introduction. From forall z: W(z) we can infer forall z: W(z) or T_4(z) via universal addition. From forall z: Q_1(z) <-> W(z), exists z: ~Q_1(z) we can infer exists z: ~W(z) via existential biconditional elimination. Finally, from forall z: W(z) or T_4(z), exists z: ~W(z) we can infer exists z: T_4(z) via existential disjunctive syllogism. Therefore, the answer is yes. Notice, however, that the premises were contradictory, so we can infer anything from them.