Consider the following premises. exists x: R(x) or ~T_3(x). forall x: T_3(x) <-> U(x). forall x: ~R(x). Can we infer exists x: ~U(x) from them? If possible, name the inference rules being used at each step.
 From exists x: R(x) or ~T_3(x) we can infer exists x: T_3(x) -> R(x) via existential rewriting. From exists x: T_3(x) -> R(x), forall x: ~R(x) we can infer exists x: ~T_3(x) via existential modus tollens. Finally, from forall x: T_3(x) <-> U(x), exists x: ~T_3(x) we can infer exists x: ~U(x) via existential biconditional elimination. Therefore, the answer is yes.