Consider the following premises. forall x_15: P_15(x_15) or Q_15(x_15). forall x_15: W(x_15) or ~S(x_15). exists x_15: P_15(x_15) -> R(x_15). forall x_15: Q_15(x_15) -> R(x_15). forall x_15: S(x_15). Can we infer exists x_15: W(x_15) and R(x_15) from them? If possible, name the inference rules being used at each step.
 From exists x_15: P_15(x_15) -> R(x_15), forall x_15: Q_15(x_15) -> R(x_15), forall x_15: P_15(x_15) or Q_15(x_15) we can infer exists x_15: R(x_15) via existential disjunction elimination. From forall x_15: W(x_15) or ~S(x_15) we can infer forall x_15: S(x_15) -> W(x_15) via universal rewriting. From forall x_15: S(x_15) -> W(x_15), forall x_15: S(x_15) we can infer forall x_15: W(x_15) via universal modus ponens. Finally, from forall x_15: W(x_15), exists x_15: R(x_15) we can infer exists x_15: W(x_15) and R(x_15) via existential conjunction. Therefore, the answer is yes.