Consider the following premises. forall x_10: ~Q_10(x_10). exists x_10: S_1(x_10) <-> Q_8(x_10). forall x_10: P_10(x_10) or S_1(x_10). forall x_10: P_10(x_10) -> Q_10(x_10). Can we infer exists x_10: Q_8(x_10) from them? If possible, name the inference rules being used at each step.
 From forall x_10: P_10(x_10) -> Q_10(x_10), forall x_10: ~Q_10(x_10) we can infer forall x_10: ~P_10(x_10) via universal modus tollens. From exists x_10: S_1(x_10) <-> Q_8(x_10) we can infer exists x_10: S_1(x_10) -> Q_8(x_10) via existential biconditional elimination. From forall x_10: P_10(x_10) or S_1(x_10), forall x_10: ~P_10(x_10) we can infer forall x_10: S_1(x_10) via universal disjunctive syllogism. Finally, from exists x_10: S_1(x_10) -> Q_8(x_10), forall x_10: S_1(x_10) we can infer exists x_10: Q_8(x_10) via existential modus ponens. Therefore, the answer is yes.