Consider the following premises. forall x: U_4(x) -> T(x). forall x: ~S(x). forall x: T(x) -> U_4(x). forall x: S(x) <-> U_4(x). Can we infer exists x: T(x) from them? If possible, name the inference rules being used at each step.
 From forall x: S(x) <-> U_4(x), forall x: ~S(x) we can infer forall x: ~U_4(x) via universal biconditional elimination. From forall x: U_4(x) -> T(x), forall x: T(x) -> U_4(x) we can infer forall x: U_4(x) <-> T(x) via universal biconditional introduction. Finally, from forall x: U_4(x) <-> T(x), forall x: ~U_4(x) we can infer forall x: ~T(x) via universal biconditional elimination, which contradicts exists x: T(x). Therefore, the answer is no.