Consider the following premises. exists x: Q_2(x) -> W(x). forall x: Q_3(x). forall x: Q_3(x) -> W(x). forall x: W(x) -> Q_3(x). Can we infer exists x: W(x) from them? If possible, name the inference rules being used at each step.
 From forall x: Q_3(x) -> W(x), forall x: W(x) -> Q_3(x) we can infer forall x: Q_3(x) <-> W(x) via universal biconditional introduction. From forall x: Q_3(x) <-> W(x) we can infer forall x: Q_3(x) -> W(x) via universal biconditional elimination. From forall x: Q_3(x) we can infer forall x: Q_3(x) or Q_2(x) via universal addition. Finally, from forall x: Q_3(x) -> W(x), exists x: Q_2(x) -> W(x), forall x: Q_3(x) or Q_2(x) we can infer exists x: W(x) via existential disjunction elimination. Therefore, the answer is yes.