Consider the following premises. forall x_15: ~R_8(x_15). exists x_15: W(x_15). forall x_15: P_15(x_15) -> R_8(x_15). forall x_15: P_15(x_15) <-> W(x_15). Can we infer exists x_15: Q_5(x_15) from them? If possible, name the inference rules being used at each step.
 From forall x_15: P_15(x_15) <-> W(x_15) we can infer forall x_15: W(x_15) -> P_15(x_15) via universal biconditional elimination. From forall x_15: W(x_15) -> P_15(x_15), forall x_15: P_15(x_15) -> R_8(x_15) we can infer forall x_15: W(x_15) -> R_8(x_15) via universal transitivity. From exists x_15: W(x_15) we can infer exists x_15: W(x_15) or Q_5(x_15) via existential addition. From forall x_15: W(x_15) -> R_8(x_15), forall x_15: ~R_8(x_15) we can infer forall x_15: ~W(x_15) via universal modus tollens. Finally, from exists x_15: W(x_15) or Q_5(x_15), forall x_15: ~W(x_15) we can infer exists x_15: Q_5(x_15) via existential disjunctive syllogism. Therefore, the answer is yes.