Consider the following premises. forall x: Q(x) or ~Q_3(x). forall x: Q_3(x) or T(x). exists x: Q(x) or ~T(x). Can we infer forall x: ~Q(x) from them? If possible, name the inference rules being used at each step.
 From exists x: Q(x) or ~T(x) we can infer exists x: T(x) -> Q(x) via existential rewriting. From forall x: Q(x) or ~Q_3(x) we can infer forall x: Q_3(x) -> Q(x) via universal rewriting. Finally, from forall x: Q_3(x) -> Q(x), exists x: T(x) -> Q(x), forall x: Q_3(x) or T(x) we can infer exists x: Q(x) via existential disjunction elimination, which contradicts forall x: ~Q(x). Therefore, the answer is no.