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