Consider the following premises. forall x: U(x) -> U_4(x). forall x: T(x) -> S(x). forall x: T(x) <-> U_4(x). exists x: S(x) -> U(x). Can we infer exists x: U(x) -> T(x) from them?
 From forall x: T(x) <-> U_4(x) we can infer forall x: U_4(x) -> T(x). From forall x: T(x) -> S(x), exists x: S(x) -> U(x) we can infer exists x: T(x) -> U(x). From forall x: U(x) -> U_4(x), forall x: U_4(x) -> T(x) we can infer forall x: U(x) -> T(x). From forall x: U(x) -> T(x), exists x: T(x) -> U(x) we can infer exists x: U(x) <-> T(x). Finally, from exists x: U(x) <-> T(x) we can infer exists x: U(x) -> T(x). Therefore, the answer is yes.