Consider the following premises. forall x_6: P_6(x_6) or P_1(x_6). forall x_6: ~P_6(x_6). forall x_6: ~U_1(x_6). forall x_6: U_1(x_6) or Q(x_6). Can we infer exists x_6: ~(P_1(x_6) and Q(x_6)) from them? If possible, name the inference rules being used at each step.
 From forall x_6: P_6(x_6) or P_1(x_6), forall x_6: ~P_6(x_6) we can infer forall x_6: P_1(x_6) via universal disjunctive syllogism. From forall x_6: U_1(x_6) or Q(x_6), forall x_6: ~U_1(x_6) we can infer forall x_6: Q(x_6) via universal disjunctive syllogism. Finally, from forall x_6: P_1(x_6), forall x_6: Q(x_6) we can infer forall x_6: P_1(x_6) and Q(x_6) via universal conjunction, which contradicts exists x_6: ~(P_1(x_6) and Q(x_6)). Therefore, the answer is no.