Consider the following premises. forall z_4: R(z_4) or ~U(z_4). forall z_4: R(z_4) -> U(z_4). forall z_4: U(z_4) -> R(z_4). Can we infer exists z_4: ~(R(z_4) -> U(z_4)) from them? If possible, name the inference rules being used at each step.
 From forall z_4: R(z_4) -> U(z_4), forall z_4: U(z_4) -> R(z_4) we can infer forall z_4: R(z_4) <-> U(z_4) via universal biconditional introduction. From forall z_4: R(z_4) <-> U(z_4) we can infer forall z_4: R(z_4) -> U(z_4) via universal biconditional elimination. From forall z_4: R(z_4) or ~U(z_4) we can infer forall z_4: U(z_4) -> R(z_4) via universal rewriting. From forall z_4: R(z_4) -> U(z_4), forall z_4: U(z_4) -> R(z_4) we can infer forall z_4: R(z_4) <-> U(z_4) via universal biconditional introduction. Finally, from forall z_4: R(z_4) <-> U(z_4) we can infer forall z_4: R(z_4) -> U(z_4) via universal biconditional elimination, which contradicts exists z_4: ~(R(z_4) -> U(z_4)). Therefore, the answer is no.