Consider the following premises. exists y: S_1(y) <-> S(y). forall y: S_1(y) <-> S(y). Can we infer forall y: ~(S(y) -> S_1(y)) from them? If possible, name the inference rules being used at each step.
 From exists y: S_1(y) <-> S(y) we can infer exists y: S_1(y) -> S(y) via existential biconditional elimination. Finally, from forall y: S_1(y) <-> S(y) we can infer forall y: S(y) -> S_1(y) via universal biconditional elimination, which contradicts forall y: ~(S(y) -> S_1(y)). Therefore, the answer is no.