Consider the following premises. exists x_15: R_4(x_15) -> P_4(x_15). forall x_15: Q_15(x_15) -> P(x_15). forall x_15: S_4(x_15) -> P_4(x_15). forall x_15: W(x_15) -> P(x_15). forall x_15: W(x_15) or Q_15(x_15). forall x_15: R_4(x_15) or S_4(x_15). Can we infer exists x_15: P_4(x_15) from them? If possible, name the inference rules being used at each step.
 From forall x_15: W(x_15) -> P(x_15), forall x_15: Q_15(x_15) -> P(x_15), forall x_15: W(x_15) or Q_15(x_15) we can infer forall x_15: P(x_15) via universal disjunction elimination. Finally, from exists x_15: R_4(x_15) -> P_4(x_15), forall x_15: S_4(x_15) -> P_4(x_15), forall x_15: R_4(x_15) or S_4(x_15) we can infer exists x_15: P_4(x_15) via existential disjunction elimination. Therefore, the answer is yes.