Consider the following premises. forall z: ~W(z). forall z: T_4(z) -> U(z). forall z: U(z) <-> T_4(z). forall z: W(z) <-> T_4(z). Can we infer exists z: U(z) from them? If possible, name the inference rules being used at each step.
 From forall z: U(z) <-> T_4(z) we can infer forall z: U(z) -> T_4(z) via universal biconditional elimination. From forall z: W(z) <-> T_4(z), forall z: ~W(z) we can infer forall z: ~T_4(z) via universal biconditional elimination. From forall z: T_4(z) -> U(z), forall z: U(z) -> T_4(z) we can infer forall z: T_4(z) <-> U(z) via universal biconditional introduction. Finally, from forall z: T_4(z) <-> U(z), forall z: ~T_4(z) we can infer forall z: ~U(z) via universal biconditional elimination, which contradicts exists z: U(z). Therefore, the answer is no.