Consider the following premises. forall z: U_4(z) -> R_4(z). forall z: R_4(z) -> S(z). forall z: S(z) -> U_1(z). exists z: U_1(z) -> U_4(z). Can we infer exists z: R_4(z) -> S(z) from them? If possible, name the inference rules being used at each step.
 From forall z: S(z) -> U_1(z), exists z: U_1(z) -> U_4(z) we can infer exists z: S(z) -> U_4(z) via existential transitivity. From exists z: S(z) -> U_4(z), forall z: U_4(z) -> R_4(z) we can infer exists z: S(z) -> R_4(z) via existential transitivity. From forall z: R_4(z) -> S(z), exists z: S(z) -> R_4(z) we can infer exists z: R_4(z) <-> S(z) via existential biconditional introduction. Finally, from exists z: R_4(z) <-> S(z) we can infer exists z: R_4(z) -> S(z) via existential biconditional elimination. Therefore, the answer is yes.