Consider the following premises. forall x_13: P_6(x_13) or U(x_13). forall x_13: ~P_6(x_13) or W_1(x_13). forall x_13: W_1(x_13) -> Q_7(x_13). forall x_13: Q_7(x_13) -> R_4(x_13). forall x_13: P_13(x_13) <-> R_4(x_13). forall x_13: U(x_13) -> P_13(x_13). Can we infer forall x_13: R_4(x_13) from them? If possible, name the inference rules being used at each step.
 From forall x_13: P_13(x_13) <-> R_4(x_13) we can infer forall x_13: P_13(x_13) -> R_4(x_13) via universal biconditional elimination. From forall x_13: W_1(x_13) -> Q_7(x_13), forall x_13: Q_7(x_13) -> R_4(x_13) we can infer forall x_13: W_1(x_13) -> R_4(x_13) via universal transitivity. From forall x_13: P_6(x_13) or U(x_13), forall x_13: ~P_6(x_13) or W_1(x_13) we can infer forall x_13: U(x_13) or W_1(x_13) via universal resolution. From forall x_13: U(x_13) -> P_13(x_13), forall x_13: P_13(x_13) -> R_4(x_13) we can infer forall x_13: U(x_13) -> R_4(x_13) via universal transitivity. Finally, from forall x_13: U(x_13) -> R_4(x_13), forall x_13: W_1(x_13) -> R_4(x_13), forall x_13: U(x_13) or W_1(x_13) we can infer forall x_13: R_4(x_13) via universal disjunction elimination. Therefore, the answer is yes.