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