Consider the following premises. forall z_3: P_1(z_3) -> R(z_3). forall z_3: R(z_3) <-> U(z_3). forall z_3: P_1(z_3) <-> R(z_3). forall z_3: ~P_1(z_3). Can we infer exists z_3: U(z_3) from them?
 From forall z_3: P_1(z_3) <-> R(z_3) we can infer forall z_3: R(z_3) -> P_1(z_3). From forall z_3: R(z_3) -> P_1(z_3), forall z_3: P_1(z_3) -> R(z_3) we can infer forall z_3: R(z_3) <-> P_1(z_3). From forall z_3: R(z_3) <-> P_1(z_3), forall z_3: ~P_1(z_3) we can infer forall z_3: ~R(z_3). Finally, from forall z_3: R(z_3) <-> U(z_3), forall z_3: ~R(z_3) we can infer forall z_3: ~U(z_3), which contradicts exists z_3: U(z_3). Therefore, the answer is no.