Consider the following premises. forall x_14: P(x_14) -> P_14(x_14). forall x_14: ~P_6(x_14) or S(x_14). forall x_14: ~Q_14(x_14). forall x_14: P_14(x_14) <-> Q_14(x_14). forall x_14: P_6(x_14) or P(x_14). Can we infer forall x_14: S(x_14) or W(x_14) from them? If possible, name the inference rules being used at each step.
 From forall x_14: P_14(x_14) <-> Q_14(x_14), forall x_14: ~Q_14(x_14) we can infer forall x_14: ~P_14(x_14) via universal biconditional elimination. From forall x_14: P(x_14) -> P_14(x_14), forall x_14: ~P_14(x_14) we can infer forall x_14: ~P(x_14) via universal modus tollens. From forall x_14: P_6(x_14) or P(x_14), forall x_14: ~P_6(x_14) or S(x_14) we can infer forall x_14: P(x_14) or S(x_14) via universal resolution. From forall x_14: P(x_14) or S(x_14), forall x_14: ~P(x_14) we can infer forall x_14: S(x_14) via universal disjunctive syllogism. Finally, from forall x_14: S(x_14) we can infer forall x_14: S(x_14) or W(x_14) via universal addition. Therefore, the answer is yes.