Consider the following premises. forall x_11: U_2(x_11). forall x_11: P_11(x_11) -> U_2(x_11). exists x_11: ~P_11(x_11). forall x_11: U_2(x_11) -> P_11(x_11). Can we infer exists x_11: Q_9(x_11) from them? If possible, name the inference rules being used at each step.
 From forall x_11: P_11(x_11) -> U_2(x_11), forall x_11: U_2(x_11) -> P_11(x_11) we can infer forall x_11: P_11(x_11) <-> U_2(x_11) via universal biconditional introduction. From forall x_11: U_2(x_11) we can infer forall x_11: U_2(x_11) or Q_9(x_11) via universal addition. From forall x_11: P_11(x_11) <-> U_2(x_11), exists x_11: ~P_11(x_11) we can infer exists x_11: ~U_2(x_11) via existential biconditional elimination. Finally, from forall x_11: U_2(x_11) or Q_9(x_11), exists x_11: ~U_2(x_11) we can infer exists x_11: Q_9(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.