Consider the following premises. forall y_4: S_3(y_4) <-> S(y_4). forall y_4: S_3(y_4) -> R_4(y_4). forall y_4: S(y_4) or ~R_4(y_4). Can we infer forall y_4: S(y_4) <-> S_3(y_4) from them? If possible, name the inference rules being used at each step.
 From forall y_4: S(y_4) or ~R_4(y_4) we can infer forall y_4: R_4(y_4) -> S(y_4) via universal rewriting. From forall y_4: S_3(y_4) -> R_4(y_4), forall y_4: R_4(y_4) -> S(y_4) we can infer forall y_4: S_3(y_4) -> S(y_4) via universal transitivity. From forall y_4: S_3(y_4) <-> S(y_4) we can infer forall y_4: S(y_4) -> S_3(y_4) via universal biconditional elimination. Finally, from forall y_4: S(y_4) -> S_3(y_4), forall y_4: S_3(y_4) -> S(y_4) we can infer forall y_4: S(y_4) <-> S_3(y_4) via universal biconditional introduction. Therefore, the answer is yes.