Consider the following premises. forall z: T_4(z) or ~S_2(z). forall z: T_4(z) -> S_2(z). forall z: S_2(z) -> T_4(z). Can we infer exists z: ~(T_4(z) <-> S_2(z)) from them?
 From forall z: S_2(z) -> T_4(z), forall z: T_4(z) -> S_2(z) we can infer forall z: S_2(z) <-> T_4(z). From forall z: S_2(z) <-> T_4(z) we can infer forall z: T_4(z) -> S_2(z). From forall z: T_4(z) or ~S_2(z) we can infer forall z: S_2(z) -> T_4(z). Finally, from forall z: T_4(z) -> S_2(z), forall z: S_2(z) -> T_4(z) we can infer forall z: T_4(z) <-> S_2(z), which contradicts exists z: ~(T_4(z) <-> S_2(z)). Therefore, the answer is no.