Consider the following premises. forall x_9: Q_9(x_9) -> T(x_9). forall x_9: ~W(x_9). forall x_9: W(x_9) <-> T(x_9). forall x_9: Q(x_9) -> Q_9(x_9). exists x_9: U(x_9) <-> Q(x_9). Can we infer exists x_9: ~U(x_9) from them? If possible, name the inference rules being used at each step.
 From forall x_9: Q(x_9) -> Q_9(x_9), forall x_9: Q_9(x_9) -> T(x_9) we can infer forall x_9: Q(x_9) -> T(x_9) via universal transitivity. From forall x_9: W(x_9) <-> T(x_9), forall x_9: ~W(x_9) we can infer forall x_9: ~T(x_9) via universal biconditional elimination. From forall x_9: Q(x_9) -> T(x_9), forall x_9: ~T(x_9) we can infer forall x_9: ~Q(x_9) via universal modus tollens. Finally, from exists x_9: U(x_9) <-> Q(x_9), forall x_9: ~Q(x_9) we can infer exists x_9: ~U(x_9) via existential biconditional elimination. Therefore, the answer is yes.