Consider the following premises. forall z: W(z) -> T_3(z). forall z: P(z) -> W(z). forall z: T_3(z) -> T(z). forall z: T_3(z) or ~T(z). exists z: P(z) or T(z). Can we infer exists z: T_3(z) from them? If possible, name the inference rules being used at each step.
 From forall z: T_3(z) or ~T(z) we can infer forall z: T(z) -> T_3(z) via universal rewriting. From forall z: P(z) -> W(z), forall z: W(z) -> T_3(z) we can infer forall z: P(z) -> T_3(z) via universal transitivity. From forall z: T(z) -> T_3(z), forall z: T_3(z) -> T(z) we can infer forall z: T(z) <-> T_3(z) via universal biconditional introduction. From forall z: T(z) <-> T_3(z) we can infer forall z: T(z) -> T_3(z) via universal biconditional elimination. Finally, from forall z: P(z) -> T_3(z), forall z: T(z) -> T_3(z), exists z: P(z) or T(z) we can infer exists z: T_3(z) via existential disjunction elimination. Therefore, the answer is yes.