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