Consider the following premises. forall x_15: Q_9(x_15) -> R_2(x_15). forall x_15: P_9(x_15) -> Q_9(x_15). forall x_15: P_9(x_15) or Q_8(x_15). forall x_15: R_2(x_15) -> P(x_15). forall x_15: Q_8(x_15) -> R_2(x_15). forall x_15: P(x_15) <-> U(x_15). Can we infer forall x_15: U(x_15) from them? If possible, name the inference rules being used at each step.
 From forall x_15: P(x_15) <-> U(x_15) we can infer forall x_15: P(x_15) -> U(x_15) via universal biconditional elimination. From forall x_15: P_9(x_15) -> Q_9(x_15), forall x_15: Q_9(x_15) -> R_2(x_15) we can infer forall x_15: P_9(x_15) -> R_2(x_15) via universal transitivity. From forall x_15: P_9(x_15) -> R_2(x_15), forall x_15: Q_8(x_15) -> R_2(x_15), forall x_15: P_9(x_15) or Q_8(x_15) we can infer forall x_15: R_2(x_15) via universal disjunction elimination. From forall x_15: R_2(x_15) -> P(x_15), forall x_15: R_2(x_15) we can infer forall x_15: P(x_15) via universal modus ponens. Finally, from forall x_15: P(x_15) -> U(x_15), forall x_15: P(x_15) we can infer forall x_15: U(x_15) via universal modus ponens. Therefore, the answer is yes.