Consider the following premises. forall z: T_3(z) -> T_4(z). exists z: U(z) or T_3(z). forall z: T_4(z) -> P_4(z). forall z: U(z) -> P_4(z). Can we infer forall z: ~P_4(z) from them? If possible, name the inference rules being used at each step.
 From forall z: T_3(z) -> T_4(z), forall z: T_4(z) -> P_4(z) we can infer forall z: T_3(z) -> P_4(z) via universal transitivity. Finally, from forall z: U(z) -> P_4(z), forall z: T_3(z) -> P_4(z), exists z: U(z) or T_3(z) we can infer exists z: P_4(z) via existential disjunction elimination, which contradicts forall z: ~P_4(z). Therefore, the answer is no.