Consider the following premises. forall y: T(y) -> S_2(y). forall y: S_2(y) <-> T_3(y). forall y: T(y) <-> T_3(y). Can we infer exists y: ~(T_3(y) -> S_2(y)) from them? If possible, name the inference rules being used at each step.
 From forall y: T(y) <-> T_3(y) we can infer forall y: T_3(y) -> T(y) via universal biconditional elimination. From forall y: T_3(y) -> T(y), forall y: T(y) -> S_2(y) we can infer forall y: T_3(y) -> S_2(y) via universal transitivity. From forall y: S_2(y) <-> T_3(y) we can infer forall y: S_2(y) -> T_3(y) via universal biconditional elimination. From forall y: S_2(y) -> T_3(y), forall y: T_3(y) -> S_2(y) we can infer forall y: S_2(y) <-> T_3(y) via universal biconditional introduction. Finally, from forall y: S_2(y) <-> T_3(y) we can infer forall y: T_3(y) -> S_2(y) via universal biconditional elimination, which contradicts exists y: ~(T_3(y) -> S_2(y)). Therefore, the answer is no.