Consider the following premises. exists z: Q(z) or T(z). forall z: U_3(z) or ~T(z). forall z: U_3(z) <-> Q(z). Can we infer exists z: U_3(z) from them? If possible, name the inference rules being used at each step.
 From forall z: U_3(z) <-> Q(z) we can infer forall z: Q(z) -> U_3(z) via universal biconditional elimination. From forall z: U_3(z) or ~T(z) we can infer forall z: T(z) -> U_3(z) via universal rewriting. Finally, from forall z: Q(z) -> U_3(z), forall z: T(z) -> U_3(z), exists z: Q(z) or T(z) we can infer exists z: U_3(z) via existential disjunction elimination. Therefore, the answer is yes.