Consider the following premises. forall x_9: S_2(x_9) -> U(x_9). forall x_9: ~P_9(x_9). forall x_9: P_9(x_9) <-> U(x_9). forall x_9: P_9(x_9) -> U(x_9). Can we infer forall x_9: ~S_2(x_9) from them? If possible, name the inference rules being used at each step.
 From forall x_9: P_9(x_9) <-> U(x_9) we can infer forall x_9: U(x_9) -> P_9(x_9) via universal biconditional elimination. From forall x_9: P_9(x_9) -> U(x_9), forall x_9: U(x_9) -> P_9(x_9) we can infer forall x_9: P_9(x_9) <-> U(x_9) via universal biconditional introduction. From forall x_9: P_9(x_9) <-> U(x_9), forall x_9: ~P_9(x_9) we can infer forall x_9: ~U(x_9) via universal biconditional elimination. Finally, from forall x_9: S_2(x_9) -> U(x_9), forall x_9: ~U(x_9) we can infer forall x_9: ~S_2(x_9) via universal modus tollens. Therefore, the answer is yes.