Consider the following premises. forall x: T_2(x) -> U(x). forall x: T_1(x) -> T_2(x). exists x: U(x) -> T_1(x). forall x: ~T_1(x). Can we infer exists x: ~U(x) from them?
 From forall x: T_1(x) -> T_2(x), forall x: T_2(x) -> U(x) we can infer forall x: T_1(x) -> U(x). From forall x: T_1(x) -> U(x), exists x: U(x) -> T_1(x) we can infer exists x: T_1(x) <-> U(x). Finally, from exists x: T_1(x) <-> U(x), forall x: ~T_1(x) we can infer exists x: ~U(x). Therefore, the answer is yes.