Consider the following premises. forall x_9: P_9(x_9) -> R_9(x_9). forall x_9: R_9(x_9) -> U(x_9). forall x_9: Q_9(x_9) -> R_9(x_9). forall x_9: P_9(x_9) or Q_9(x_9). forall x_9: W_2(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: P_9(x_9) -> R_9(x_9), forall x_9: Q_9(x_9) -> R_9(x_9), forall x_9: P_9(x_9) or Q_9(x_9) we can infer forall x_9: R_9(x_9) via universal disjunction elimination. From forall x_9: R_9(x_9) -> U(x_9), forall x_9: R_9(x_9) we can infer forall x_9: U(x_9) via universal modus ponens. From forall x_9: W_2(x_9), forall x_9: U(x_9) we can infer forall x_9: W_2(x_9) and U(x_9) via universal conjunction. Finally, from forall x_9: W_2(x_9) and U(x_9) we can infer forall x_9: W_2(x_9) via universal simplification, which contradicts exists x_9: ~U(x_9). Therefore, the answer is no.