Consider the following premises. exists z_2: P(z_2) <-> U_4(z_2). forall z_2: U_4(z_2) or ~P(z_2). Can we infer exists z_2: U_4(z_2) <-> P(z_2) from them?
 From forall z_2: U_4(z_2) or ~P(z_2) we can infer forall z_2: P(z_2) -> U_4(z_2). From exists z_2: P(z_2) <-> U_4(z_2) we can infer exists z_2: U_4(z_2) -> P(z_2). Finally, from exists z_2: U_4(z_2) -> P(z_2), forall z_2: P(z_2) -> U_4(z_2) we can infer exists z_2: U_4(z_2) <-> P(z_2). Therefore, the answer is yes.