Consider the following premises. forall z_2: U(z_2) or ~T(z_2). exists z_2: U(z_2). forall z_2: U(z_2) -> T(z_2). Can we infer exists z_2: T(z_2) from them? If possible, name the inference rules being used at each step.
 From forall z_2: U(z_2) or ~T(z_2) we can infer forall z_2: T(z_2) -> U(z_2) via universal rewriting. From forall z_2: T(z_2) -> U(z_2), forall z_2: U(z_2) -> T(z_2) we can infer forall z_2: T(z_2) <-> U(z_2) via universal biconditional introduction. From forall z_2: T(z_2) <-> U(z_2) we can infer forall z_2: U(z_2) -> T(z_2) via universal biconditional elimination. Finally, from forall z_2: U(z_2) -> T(z_2), exists z_2: U(z_2) we can infer exists z_2: T(z_2) via existential modus ponens. Therefore, the answer is yes.