Consider the following premises. forall x_5: P_2(x_5) or P(x_5). exists x_5: W_4(x_5) or ~R_4(x_5). forall x_5: ~P_2(x_5) or R_4(x_5). forall x_5: P(x_5) -> W_4(x_5). Can we infer exists x_5: W_4(x_5) from them? If possible, name the inference rules being used at each step.
 From exists x_5: W_4(x_5) or ~R_4(x_5) we can infer exists x_5: R_4(x_5) -> W_4(x_5) via existential rewriting. From forall x_5: P_2(x_5) or P(x_5), forall x_5: ~P_2(x_5) or R_4(x_5) we can infer forall x_5: P(x_5) or R_4(x_5) via universal resolution. Finally, from forall x_5: P(x_5) -> W_4(x_5), exists x_5: R_4(x_5) -> W_4(x_5), forall x_5: P(x_5) or R_4(x_5) we can infer exists x_5: W_4(x_5) via existential disjunction elimination. Therefore, the answer is yes.