Consider the following premises. forall z: T(z) or W(z). exists z: Q_4(z) or ~U_3(z). forall z: T(z) -> Q_4(z). forall z: W(z) -> U_3(z). Can we infer forall z: ~Q_4(z) from them? If possible, name the inference rules being used at each step.
 From exists z: Q_4(z) or ~U_3(z) we can infer exists z: U_3(z) -> Q_4(z) via existential rewriting. From forall z: W(z) -> U_3(z), exists z: U_3(z) -> Q_4(z) we can infer exists z: W(z) -> Q_4(z) via existential transitivity. Finally, from forall z: T(z) -> Q_4(z), exists z: W(z) -> Q_4(z), forall z: T(z) or W(z) we can infer exists z: Q_4(z) via existential disjunction elimination, which contradicts forall z: ~Q_4(z). Therefore, the answer is no.