Consider the following premises. forall x_13: P_4(x_13) <-> S(x_13). forall x_13: ~P_4(x_13). forall x_13: S(x_13) <-> W(x_13). forall x_13: W(x_13) or ~S(x_13). Can we infer forall x_13: ~W(x_13) from them? If possible, name the inference rules being used at each step.
 From forall x_13: S(x_13) <-> W(x_13) we can infer forall x_13: W(x_13) -> S(x_13) via universal biconditional elimination. From forall x_13: W(x_13) or ~S(x_13) we can infer forall x_13: S(x_13) -> W(x_13) via universal rewriting. From forall x_13: P_4(x_13) <-> S(x_13), forall x_13: ~P_4(x_13) we can infer forall x_13: ~S(x_13) via universal biconditional elimination. From forall x_13: S(x_13) -> W(x_13), forall x_13: W(x_13) -> S(x_13) we can infer forall x_13: S(x_13) <-> W(x_13) via universal biconditional introduction. Finally, from forall x_13: S(x_13) <-> W(x_13), forall x_13: ~S(x_13) we can infer forall x_13: ~W(x_13) via universal biconditional elimination. Therefore, the answer is yes.