Consider the following premises. forall z: U(z) -> P_3(z). forall z: ~T(z). forall z: P_3(z) <-> T(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_3(z) <-> T(z) we can infer forall z: P_3(z) -> T(z) via universal biconditional elimination. From forall z: U(z) -> P_3(z), forall z: P_3(z) -> T(z) we can infer forall z: U(z) -> T(z) via universal transitivity. Finally, from forall z: U(z) -> T(z), forall z: ~T(z) we can infer forall z: ~U(z) via universal modus tollens. Therefore, the answer is yes.