Consider the following premises. forall x_8: P_4(x_8) or S(x_8). exists x_8: P_8(x_8) -> Q_8(x_8). forall x_8: P_8(x_8). forall x_8: ~P_4(x_8). Can we infer exists x_8: S(x_8) from them? If possible, name the inference rules being used at each step.
 From exists x_8: P_8(x_8) -> Q_8(x_8), forall x_8: P_8(x_8) we can infer exists x_8: Q_8(x_8) via existential modus ponens. From forall x_8: P_4(x_8) or S(x_8), forall x_8: ~P_4(x_8) we can infer forall x_8: S(x_8) via universal disjunctive syllogism. From forall x_8: S(x_8), exists x_8: Q_8(x_8) we can infer exists x_8: S(x_8) and Q_8(x_8) via existential conjunction. Finally, from exists x_8: S(x_8) and Q_8(x_8) we can infer exists x_8: S(x_8) via existential simplification. Therefore, the answer is yes.