Consider the following premises. forall x_10: Q_10(x_10) -> P_1(x_10). forall x_10: U(x_10) -> Q_10(x_10). forall x_10: P_1(x_10) -> U(x_10). forall x_10: P_1(x_10) -> Q_10(x_10). Can we infer forall x_10: U(x_10) -> P_1(x_10) from them? If possible, name the inference rules being used at each step.
 From forall x_10: P_1(x_10) -> Q_10(x_10), forall x_10: Q_10(x_10) -> P_1(x_10) we can infer forall x_10: P_1(x_10) <-> Q_10(x_10) via universal biconditional introduction. From forall x_10: P_1(x_10) <-> Q_10(x_10) we can infer forall x_10: Q_10(x_10) -> P_1(x_10) via universal biconditional elimination. Finally, from forall x_10: U(x_10) -> Q_10(x_10), forall x_10: Q_10(x_10) -> P_1(x_10) we can infer forall x_10: U(x_10) -> P_1(x_10) via universal transitivity. Therefore, the answer is yes.