Consider the following premises. forall x_15: P_3(x_15). forall x_15: Q_15(x_15) -> R_15(x_15). exists x_15: P_15(x_15) or Q_15(x_15). forall x_15: P_3(x_15) -> R_4(x_15). forall x_15: P_15(x_15) -> R_15(x_15). Can we infer exists x_15: R_4(x_15) from them? If possible, name the inference rules being used at each step.
 From forall x_15: P_15(x_15) -> R_15(x_15), forall x_15: Q_15(x_15) -> R_15(x_15), exists x_15: P_15(x_15) or Q_15(x_15) we can infer exists x_15: R_15(x_15) via existential disjunction elimination. From forall x_15: P_3(x_15) -> R_4(x_15), forall x_15: P_3(x_15) we can infer forall x_15: R_4(x_15) via universal modus ponens. From forall x_15: R_4(x_15), exists x_15: R_15(x_15) we can infer exists x_15: R_4(x_15) and R_15(x_15) via existential conjunction. Finally, from exists x_15: R_4(x_15) and R_15(x_15) we can infer exists x_15: R_4(x_15) via existential simplification. Therefore, the answer is yes.