Consider the following premises. forall x_11: P_11(x_11) or Q_11(x_11). forall x_11: ~Q_11(x_11). forall x_11: P_4(x_11). forall x_11: ~P_11(x_11) or R_11(x_11). Can we infer exists x_11: ~(P_4(x_11) or P(x_11)) from them? If possible, name the inference rules being used at each step.
 From forall x_11: P_11(x_11) or Q_11(x_11), forall x_11: ~P_11(x_11) or R_11(x_11) we can infer forall x_11: Q_11(x_11) or R_11(x_11) via universal resolution. From forall x_11: Q_11(x_11) or R_11(x_11), forall x_11: ~Q_11(x_11) we can infer forall x_11: R_11(x_11) via universal disjunctive syllogism. From forall x_11: P_4(x_11), forall x_11: R_11(x_11) we can infer forall x_11: P_4(x_11) and R_11(x_11) via universal conjunction. From forall x_11: P_4(x_11) and R_11(x_11) we can infer forall x_11: P_4(x_11) via universal simplification. Finally, from forall x_11: P_4(x_11) we can infer forall x_11: P_4(x_11) or P(x_11) via universal addition, which contradicts exists x_11: ~(P_4(x_11) or P(x_11)). Therefore, the answer is no.