Consider the following premises. forall z: P_1(z) -> U(z). forall z: U(z) -> P_1(z). exists z: U(z) -> P_1(z). Can we infer exists z: U(z) <-> P_1(z) from them? If possible, name the inference rules being used at each step.
 From forall z: P_1(z) -> U(z), exists z: U(z) -> P_1(z) we can infer exists z: P_1(z) <-> U(z) via existential biconditional introduction. From exists z: P_1(z) <-> U(z) we can infer exists z: P_1(z) -> U(z) via existential biconditional elimination. Finally, from forall z: U(z) -> P_1(z), exists z: P_1(z) -> U(z) we can infer exists z: U(z) <-> P_1(z) via existential biconditional introduction. Therefore, the answer is yes.