Consider the following premises. forall z_1: U(z_1) or ~T(z_1). forall z_1: U(z_1) -> T(z_1). exists z_1: U(z_1) -> T(z_1). Can we infer forall z_1: ~(U(z_1) <-> T(z_1)) from them? If possible, name the inference rules being used at each step.
 From forall z_1: U(z_1) or ~T(z_1) we can infer forall z_1: T(z_1) -> U(z_1) via universal rewriting. Finally, from forall z_1: U(z_1) -> T(z_1), forall z_1: T(z_1) -> U(z_1) we can infer forall z_1: U(z_1) <-> T(z_1) via universal biconditional introduction, which contradicts forall z_1: ~(U(z_1) <-> T(z_1)). Therefore, the answer is no.