Consider the following premises. exists z_1: U(z_1) or ~P_1(z_1). forall z_1: P_1(z_1) -> U(z_1). forall z_1: U(z_1) -> P_1(z_1). Can we infer exists z_1: U(z_1) <-> P_1(z_1) from them? If possible, name the inference rules being used at each step.
 From forall z_1: U(z_1) -> P_1(z_1), forall z_1: P_1(z_1) -> U(z_1) we can infer forall z_1: U(z_1) <-> P_1(z_1) via universal biconditional introduction. From exists z_1: U(z_1) or ~P_1(z_1) we can infer exists z_1: P_1(z_1) -> U(z_1) via existential rewriting. From forall z_1: U(z_1) <-> P_1(z_1) we can infer forall z_1: U(z_1) -> P_1(z_1) via universal biconditional elimination. Finally, from forall z_1: U(z_1) -> P_1(z_1), exists z_1: P_1(z_1) -> U(z_1) we can infer exists z_1: U(z_1) <-> P_1(z_1) via existential biconditional introduction. Therefore, the answer is yes.