Consider the following premises. forall x_9: P_9(x_9). forall x_9: Q_8(x_9) or Q_4(x_9). forall x_9: S(x_9) <-> Q_8(x_9). forall x_9: P_9(x_9) -> S_1(x_9). forall x_9: Q_4(x_9) -> S(x_9). Can we infer forall x_9: S_1(x_9) and S(x_9) from them? If possible, name the inference rules being used at each step.
 From forall x_9: P_9(x_9) -> S_1(x_9), forall x_9: P_9(x_9) we can infer forall x_9: S_1(x_9) via universal modus ponens. From forall x_9: S(x_9) <-> Q_8(x_9) we can infer forall x_9: Q_8(x_9) -> S(x_9) via universal biconditional elimination. From forall x_9: Q_8(x_9) -> S(x_9), forall x_9: Q_4(x_9) -> S(x_9), forall x_9: Q_8(x_9) or Q_4(x_9) we can infer forall x_9: S(x_9) via universal disjunction elimination. Finally, from forall x_9: S_1(x_9), forall x_9: S(x_9) we can infer forall x_9: S_1(x_9) and S(x_9) via universal conjunction. Therefore, the answer is yes.