Consider the following premises. forall z: P_4(z) -> U(z). exists z: U_3(z) -> Q_1(z). forall z: ~U(z). forall z: Q_1(z) -> P_4(z). Can we infer exists z: ~U_3(z) from them? If possible, name the inference rules being used at each step.
 From forall z: Q_1(z) -> P_4(z), forall z: P_4(z) -> U(z) we can infer forall z: Q_1(z) -> U(z) via universal transitivity. From exists z: U_3(z) -> Q_1(z), forall z: Q_1(z) -> U(z) we can infer exists z: U_3(z) -> U(z) via existential transitivity. Finally, from exists z: U_3(z) -> U(z), forall z: ~U(z) we can infer exists z: ~U_3(z) via existential modus tollens. Therefore, the answer is yes.