Consider the following premises. forall x: W(x) -> Q_4(x). exists x: ~W(x). forall x: Q_4(x) <-> W(x). forall x: Q_4(x). Can we infer exists x: P_2(x) from them? If possible, name the inference rules being used at each step.
 From forall x: Q_4(x) <-> W(x) we can infer forall x: Q_4(x) -> W(x) via universal biconditional elimination. From forall x: W(x) -> Q_4(x), forall x: Q_4(x) -> W(x) we can infer forall x: W(x) <-> Q_4(x) via universal biconditional introduction. From forall x: W(x) <-> Q_4(x), exists x: ~W(x) we can infer exists x: ~Q_4(x) via existential biconditional elimination. From forall x: Q_4(x) we can infer forall x: Q_4(x) or P_2(x) via universal addition. Finally, from forall x: Q_4(x) or P_2(x), exists x: ~Q_4(x) we can infer exists x: P_2(x) via existential disjunctive syllogism. Therefore, the answer is yes. Notice, however, that the premises were contradictory, so we can infer anything from them.