Consider the following premises. forall x_10: P_6(x_10) or U(x_10). forall x_10: ~Q_10(x_10). forall x_10: ~P_6(x_10) or R_6(x_10). forall x_10: U(x_10) <-> Q_10(x_10). Can we infer ~R_6(a) from them? If possible, name the inference rules being used at each step.
 From forall x_10: U(x_10) <-> Q_10(x_10), forall x_10: ~Q_10(x_10) we can infer forall x_10: ~U(x_10) via universal biconditional elimination. From forall x_10: P_6(x_10) or U(x_10), forall x_10: ~P_6(x_10) or R_6(x_10) we can infer forall x_10: U(x_10) or R_6(x_10) via universal resolution. Finally, from forall x_10: U(x_10) or R_6(x_10), forall x_10: ~U(x_10) we can infer forall x_10: R_6(x_10) via universal disjunctive syllogism, which contradicts ~R_6(a). Therefore, the answer is no.