Consider the following premises. forall y: T(y) -> P(y). forall y: S_2(y) <-> T(y). forall y: P(y) -> S_2(y). exists y: ~P(y). Can we infer forall y: S_2(y) from them? If possible, name the inference rules being used at each step.
 From forall y: S_2(y) <-> T(y) we can infer forall y: S_2(y) -> T(y) via universal biconditional elimination. From forall y: S_2(y) -> T(y), forall y: T(y) -> P(y) we can infer forall y: S_2(y) -> P(y) via universal transitivity. From forall y: S_2(y) -> P(y), forall y: P(y) -> S_2(y) we can infer forall y: S_2(y) <-> P(y) via universal biconditional introduction. Finally, from forall y: S_2(y) <-> P(y), exists y: ~P(y) we can infer exists y: ~S_2(y) via existential biconditional elimination, which contradicts forall y: S_2(y). Therefore, the answer is no.