Consider the following premises. forall z: T_4(z) -> U(z). forall z: U(z) -> T_4(z). forall z: T_4(z) -> P_2(z). exists z: P_2(z) -> T_4(z). Can we infer exists z: U(z) -> P_2(z) from them?
 From forall z: U(z) -> T_4(z), forall z: T_4(z) -> U(z) we can infer forall z: U(z) <-> T_4(z). From forall z: T_4(z) -> P_2(z), exists z: P_2(z) -> T_4(z) we can infer exists z: T_4(z) <-> P_2(z). From exists z: T_4(z) <-> P_2(z) we can infer exists z: T_4(z) -> P_2(z). From forall z: U(z) <-> T_4(z) we can infer forall z: U(z) -> T_4(z). Finally, from forall z: U(z) -> T_4(z), exists z: T_4(z) -> P_2(z) we can infer exists z: U(z) -> P_2(z). Therefore, the answer is yes.