Consider the following premises. forall z_3: U_3(z_3) -> U(z_3). forall z_3: ~U_3(z_3). forall z_3: U(z_3) -> R_2(z_3). forall z_3: R_2(z_3) -> U_3(z_3). Can we infer exists z_3: U(z_3) from them? If possible, name the inference rules being used at each step.
 From forall z_3: U(z_3) -> R_2(z_3), forall z_3: R_2(z_3) -> U_3(z_3) we can infer forall z_3: U(z_3) -> U_3(z_3) via universal transitivity. From forall z_3: U(z_3) -> U_3(z_3), forall z_3: U_3(z_3) -> U(z_3) we can infer forall z_3: U(z_3) <-> U_3(z_3) via universal biconditional introduction. From forall z_3: U(z_3) <-> U_3(z_3) we can infer forall z_3: U(z_3) -> U_3(z_3) via universal biconditional elimination. Finally, from forall z_3: U(z_3) -> U_3(z_3), forall z_3: ~U_3(z_3) we can infer forall z_3: ~U(z_3) via universal modus tollens, which contradicts exists z_3: U(z_3). Therefore, the answer is no.