Consider the following premises. forall x: T_2(x) -> S(x). forall x: S(x) -> T_2(x). forall x: T_2(x) -> S(x). Can we infer exists x: ~(T_2(x) <-> S(x)) from them? If possible, name the inference rules being used at each step.
 From forall x: T_2(x) -> S(x), forall x: S(x) -> T_2(x) we can infer forall x: T_2(x) <-> S(x) via universal biconditional introduction. From forall x: T_2(x) <-> S(x) we can infer forall x: S(x) -> T_2(x) via universal biconditional elimination. Finally, from forall x: T_2(x) -> S(x), forall x: S(x) -> T_2(x) we can infer forall x: T_2(x) <-> S(x) via universal biconditional introduction, which contradicts exists x: ~(T_2(x) <-> S(x)). Therefore, the answer is no.