Consider the following premises. forall x_16: Q_1(x_16). forall x_16: ~R_16(x_16). forall x_16: Q_16(x_16) -> R_16(x_16). exists x_16: P_16(x_16) or W(x_16). forall x_16: P_16(x_16) -> Q_16(x_16). Can we infer exists x_16: W(x_16) from them? If possible, name the inference rules being used at each step.
 From forall x_16: P_16(x_16) -> Q_16(x_16), forall x_16: Q_16(x_16) -> R_16(x_16) we can infer forall x_16: P_16(x_16) -> R_16(x_16) via universal transitivity. From forall x_16: P_16(x_16) -> R_16(x_16), forall x_16: ~R_16(x_16) we can infer forall x_16: ~P_16(x_16) via universal modus tollens. Finally, from exists x_16: P_16(x_16) or W(x_16), forall x_16: ~P_16(x_16) we can infer exists x_16: W(x_16) via existential disjunctive syllogism. Therefore, the answer is yes.