Consider the following premises. forall z: P_4(z) or ~P_1(z). forall z: U(z) -> P_1(z). forall z: P_4(z) -> U(z). Can we infer forall z: U(z) <-> P_1(z) from them?
 From forall z: P_4(z) or ~P_1(z) we can infer forall z: P_1(z) -> P_4(z). From forall z: P_1(z) -> P_4(z), forall z: P_4(z) -> U(z) we can infer forall z: P_1(z) -> U(z). Finally, from forall z: U(z) -> P_1(z), forall z: P_1(z) -> U(z) we can infer forall z: U(z) <-> P_1(z). Therefore, the answer is yes.