Consider the following premises. forall x_10: P_10(x_10) or P(x_10). forall x_10: P(x_10) -> Q_4(x_10). forall x_10: ~P_10(x_10). forall x_10: Q_4(x_10) <-> S(x_10). Can we infer forall x_10: S(x_10) from them? If possible, name the inference rules being used at each step.
 From forall x_10: P_10(x_10) or P(x_10), forall x_10: ~P_10(x_10) we can infer forall x_10: P(x_10) via universal disjunctive syllogism. From forall x_10: P(x_10) -> Q_4(x_10), forall x_10: P(x_10) we can infer forall x_10: Q_4(x_10) via universal modus ponens. From forall x_10: Q_4(x_10) <-> S(x_10) we can infer forall x_10: Q_4(x_10) -> S(x_10) via universal biconditional elimination. Finally, from forall x_10: Q_4(x_10) -> S(x_10), forall x_10: Q_4(x_10) we can infer forall x_10: S(x_10) via universal modus ponens. Therefore, the answer is yes.