Consider the following premises. exists x: T_2(x) -> Q_4(x). forall x: Q(x) -> T_2(x). forall x: ~Q_4(x). forall x: T_2(x) -> Q(x). Can we infer exists x: ~Q(x) from them? If possible, name the inference rules being used at each step.
 From forall x: Q(x) -> T_2(x), forall x: T_2(x) -> Q(x) we can infer forall x: Q(x) <-> T_2(x) via universal biconditional introduction. From exists x: T_2(x) -> Q_4(x), forall x: ~Q_4(x) we can infer exists x: ~T_2(x) via existential modus tollens. Finally, from forall x: Q(x) <-> T_2(x), exists x: ~T_2(x) we can infer exists x: ~Q(x) via existential biconditional elimination. Therefore, the answer is yes.