Consider the following premises. forall x_15: P_4(x_15) or Q_15(x_15). exists x_15: P_8(x_15) or Q_8(x_15). forall x_15: S_3(x_15) or ~Q_15(x_15). forall x_15: P_4(x_15) -> S_3(x_15). forall x_15: ~P_8(x_15). Can we infer exists x_15: S_3(x_15) and Q_8(x_15) from them? If possible, name the inference rules being used at each step.
 From forall x_15: S_3(x_15) or ~Q_15(x_15) we can infer forall x_15: Q_15(x_15) -> S_3(x_15) via universal rewriting. From exists x_15: P_8(x_15) or Q_8(x_15), forall x_15: ~P_8(x_15) we can infer exists x_15: Q_8(x_15) via existential disjunctive syllogism. From forall x_15: P_4(x_15) -> S_3(x_15), forall x_15: Q_15(x_15) -> S_3(x_15), forall x_15: P_4(x_15) or Q_15(x_15) we can infer forall x_15: S_3(x_15) via universal disjunction elimination. Finally, from forall x_15: S_3(x_15), exists x_15: Q_8(x_15) we can infer exists x_15: S_3(x_15) and Q_8(x_15) via existential conjunction. Therefore, the answer is yes.