Consider the following premises. forall x_9: W(x_9) -> Q_9(x_9). forall x_9: Q_5(x_9) -> W(x_9). forall x_9: Q_9(x_9) -> Q_5(x_9). forall x_9: Q_9(x_9) -> W(x_9). Can we infer exists x_9: ~(W(x_9) <-> Q_9(x_9)) from them? If possible, name the inference rules being used at each step.
 From forall x_9: W(x_9) -> Q_9(x_9), forall x_9: Q_9(x_9) -> W(x_9) we can infer forall x_9: W(x_9) <-> Q_9(x_9) via universal biconditional introduction. From forall x_9: Q_9(x_9) -> Q_5(x_9), forall x_9: Q_5(x_9) -> W(x_9) we can infer forall x_9: Q_9(x_9) -> W(x_9) via universal transitivity. From forall x_9: W(x_9) <-> Q_9(x_9) we can infer forall x_9: W(x_9) -> Q_9(x_9) via universal biconditional elimination. Finally, from forall x_9: W(x_9) -> Q_9(x_9), forall x_9: Q_9(x_9) -> W(x_9) we can infer forall x_9: W(x_9) <-> Q_9(x_9) via universal biconditional introduction, which contradicts exists x_9: ~(W(x_9) <-> Q_9(x_9)). Therefore, the answer is no.