Consider the following premises. exists x_9: S(x_9). forall x_9: P_9(x_9) -> Q_9(x_9). forall x_9: R_9(x_9) -> U(x_9). forall x_9: P_9(x_9). forall x_9: Q_9(x_9) -> R_9(x_9). Can we infer forall x_9: ~(U(x_9) and S(x_9)) from them? If possible, name the inference rules being used at each step.
 From forall x_9: P_9(x_9) -> Q_9(x_9), forall x_9: Q_9(x_9) -> R_9(x_9) we can infer forall x_9: P_9(x_9) -> R_9(x_9) via universal transitivity. From forall x_9: P_9(x_9) -> R_9(x_9), forall x_9: P_9(x_9) we can infer forall x_9: R_9(x_9) via universal modus ponens. 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. Finally, from forall x_9: U(x_9), exists x_9: S(x_9) we can infer exists x_9: U(x_9) and S(x_9) via existential conjunction, which contradicts forall x_9: ~(U(x_9) and S(x_9)). Therefore, the answer is no.