Consider the following premises. forall x: R_3(x). forall x: R_3(x) <-> Q(x). exists x: ~Q(x). Can we infer exists x: Q_3(x) from them? If possible, name the inference rules being used at each step.
 From forall x: R_3(x) we can infer forall x: R_3(x) or Q_3(x) via universal addition. From forall x: R_3(x) <-> Q(x), exists x: ~Q(x) we can infer exists x: ~R_3(x) via existential biconditional elimination. Finally, from forall x: R_3(x) or Q_3(x), exists x: ~R_3(x) we can infer exists x: Q_3(x) via existential disjunctive syllogism. Therefore, the answer is yes. Notice, however, that the premises were contradictory, so we can infer anything from them.