Consider the following premises. forall z: Q_2(z). forall z: P_1(z) -> Q_2(z). forall z: U(z) -> ~P_1(z). forall z: Q_2(z) -> P_1(z). Can we infer forall z: ~U(z) from them? If possible, name the inference rules being used at each step.
 From forall z: P_1(z) -> Q_2(z), forall z: Q_2(z) -> P_1(z) we can infer forall z: P_1(z) <-> Q_2(z) via universal biconditional introduction. From forall z: P_1(z) <-> Q_2(z) we can infer forall z: Q_2(z) -> P_1(z) via universal biconditional elimination. From forall z: Q_2(z) -> P_1(z), forall z: Q_2(z) we can infer forall z: P_1(z) via universal modus ponens. Finally, from forall z: U(z) -> ~P_1(z), forall z: P_1(z) we can infer forall z: ~U(z) via universal modus tollens. Therefore, the answer is yes.