Consider the following premises. forall z: Q(z). forall z: U(z). exists z: Q(z) -> S(z). forall z: S(z) -> R_3(z). Can we infer exists z: U(z) from them? If possible, name the inference rules being used at each step.
 From exists z: Q(z) -> S(z), forall z: S(z) -> R_3(z) we can infer exists z: Q(z) -> R_3(z) via existential transitivity. From exists z: Q(z) -> R_3(z), forall z: Q(z) we can infer exists z: R_3(z) via existential modus ponens. From forall z: U(z), exists z: R_3(z) we can infer exists z: U(z) and R_3(z) via existential conjunction. Finally, from exists z: U(z) and R_3(z) we can infer exists z: U(z) via existential simplification. Therefore, the answer is yes.