Consider the following premises. forall x_9: P_8(x_9) <-> W_2(x_9). forall x_9: W_2(x_9). forall x_9: Q_9(x_9). Can we infer exists x_9: ~P_8(x_9) from them? If possible, name the inference rules being used at each step.
 From forall x_9: W_2(x_9), forall x_9: Q_9(x_9) we can infer forall x_9: W_2(x_9) and Q_9(x_9) via universal conjunction. From forall x_9: P_8(x_9) <-> W_2(x_9) we can infer forall x_9: W_2(x_9) -> P_8(x_9) via universal biconditional elimination. From forall x_9: W_2(x_9) and Q_9(x_9) we can infer forall x_9: W_2(x_9) via universal simplification. Finally, from forall x_9: W_2(x_9) -> P_8(x_9), forall x_9: W_2(x_9) we can infer forall x_9: P_8(x_9) via universal modus ponens, which contradicts exists x_9: ~P_8(x_9). Therefore, the answer is no.