Consider the following premises. forall x: W(x) -> S(x). forall x: S(x) -> W_1(x). forall x: S(x) or ~W_1(x). Can we infer forall x: W(x) -> W_1(x) from them? If possible, name the inference rules being used at each step.
 From forall x: S(x) or ~W_1(x) we can infer forall x: W_1(x) -> S(x) via universal rewriting. From forall x: W_1(x) -> S(x), forall x: S(x) -> W_1(x) we can infer forall x: W_1(x) <-> S(x) via universal biconditional introduction. From forall x: W_1(x) <-> S(x) we can infer forall x: S(x) -> W_1(x) via universal biconditional elimination. Finally, from forall x: W(x) -> S(x), forall x: S(x) -> W_1(x) we can infer forall x: W(x) -> W_1(x) via universal transitivity. Therefore, the answer is yes.