Consider the following premises. exists x_12: S(x_12). forall x_12: P_12(x_12). forall x_12: P_12(x_12) -> Q_12(x_12). Can we infer forall x_12: ~(S(x_12) or U(x_12)) from them? If possible, name the inference rules being used at each step.
 From forall x_12: P_12(x_12) -> Q_12(x_12), forall x_12: P_12(x_12) we can infer forall x_12: Q_12(x_12) via universal modus ponens. From exists x_12: S(x_12), forall x_12: Q_12(x_12) we can infer exists x_12: S(x_12) and Q_12(x_12) via existential conjunction. From exists x_12: S(x_12) and Q_12(x_12) we can infer exists x_12: S(x_12) via existential simplification. Finally, from exists x_12: S(x_12) we can infer exists x_12: S(x_12) or U(x_12) via existential addition, which contradicts forall x_12: ~(S(x_12) or U(x_12)). Therefore, the answer is no.