Consider the following premises. forall x_11: P_11(x_11) -> U_4(x_11). forall x_11: ~P_2(x_11) or R_2(x_11). forall x_11: ~P_9(x_11). forall x_11: P_9(x_11) <-> P_11(x_11). forall x_11: U_4(x_11) -> P_11(x_11). forall x_11: P_2(x_11) or U_4(x_11). Can we infer forall x_11: R_2(x_11) from them? If possible, name the inference rules being used at each step.
 From forall x_11: P_11(x_11) -> U_4(x_11), forall x_11: U_4(x_11) -> P_11(x_11) we can infer forall x_11: P_11(x_11) <-> U_4(x_11) via universal biconditional introduction. From forall x_11: P_9(x_11) <-> P_11(x_11), forall x_11: ~P_9(x_11) we can infer forall x_11: ~P_11(x_11) via universal biconditional elimination. From forall x_11: P_11(x_11) <-> U_4(x_11), forall x_11: ~P_11(x_11) we can infer forall x_11: ~U_4(x_11) via universal biconditional elimination. From forall x_11: P_2(x_11) or U_4(x_11), forall x_11: ~P_2(x_11) or R_2(x_11) we can infer forall x_11: U_4(x_11) or R_2(x_11) via universal resolution. Finally, from forall x_11: U_4(x_11) or R_2(x_11), forall x_11: ~U_4(x_11) we can infer forall x_11: R_2(x_11) via universal disjunctive syllogism. Therefore, the answer is yes.