Consider the following premises. exists z: ~U_1(z). forall z: T(z) <-> S(z). forall z: U_1(z) <-> U(z). forall z: U(z) -> S(z). forall z: S(z) -> U(z). Can we infer forall z: T(z) from them? If possible, name the inference rules being used at each step.
 From forall z: S(z) -> U(z), forall z: U(z) -> S(z) we can infer forall z: S(z) <-> U(z) via universal biconditional introduction. From forall z: U_1(z) <-> U(z), exists z: ~U_1(z) we can infer exists z: ~U(z) via existential biconditional elimination. From forall z: S(z) <-> U(z), exists z: ~U(z) we can infer exists z: ~S(z) via existential biconditional elimination. Finally, from forall z: T(z) <-> S(z), exists z: ~S(z) we can infer exists z: ~T(z) via existential biconditional elimination, which contradicts forall z: T(z). Therefore, the answer is no.