Consider the following premises. forall z: S_1(z) or ~P(z). forall z: P(z) <-> S_1(z). Can we infer forall z: S_1(z) <-> P(z) from them? If possible, name the inference rules being used at each step.
 From forall z: S_1(z) or ~P(z) we can infer forall z: P(z) -> S_1(z) via universal rewriting. From forall z: P(z) <-> S_1(z) we can infer forall z: S_1(z) -> P(z) via universal biconditional elimination. Finally, from forall z: S_1(z) -> P(z), forall z: P(z) -> S_1(z) we can infer forall z: S_1(z) <-> P(z) via universal biconditional introduction. Therefore, the answer is yes.