Consider the following premises. exists x_11: W(x_11). forall x_11: ~Q_11(x_11). forall x_11: Q_11(x_11) -> W(x_11). forall x_11: W(x_11) <-> Q_11(x_11). Can we infer exists x_11: Q_1(x_11) from them? If possible, name the inference rules being used at each step.
 From forall x_11: W(x_11) <-> Q_11(x_11) we can infer forall x_11: W(x_11) -> Q_11(x_11) via universal biconditional elimination. From forall x_11: Q_11(x_11) -> W(x_11), forall x_11: W(x_11) -> Q_11(x_11) we can infer forall x_11: Q_11(x_11) <-> W(x_11) via universal biconditional introduction. From forall x_11: Q_11(x_11) <-> W(x_11), forall x_11: ~Q_11(x_11) we can infer forall x_11: ~W(x_11) via universal biconditional elimination. From exists x_11: W(x_11) we can infer exists x_11: W(x_11) or Q_1(x_11) via existential addition. Finally, from exists x_11: W(x_11) or Q_1(x_11), forall x_11: ~W(x_11) we can infer exists x_11: Q_1(x_11) via existential disjunctive syllogism. Therefore, the answer is yes. Notice, however, that the premises were contradictory, so we can infer anything from them.