Consider the following premises. r -> P_2(j). forall z_1: P_2(z_1) -> U_2(z_1). q_4. U_2(j) or r. forall z_1: U_2(z_1) -> P_2(z_1). Can we infer ~P_2(j) from them?
 From forall z_1: U_2(z_1) -> P_2(z_1), forall z_1: P_2(z_1) -> U_2(z_1) we can infer forall z_1: U_2(z_1) <-> P_2(z_1). From forall z_1: U_2(z_1) <-> P_2(z_1) we can infer U_2(j) <-> P_2(j). From U_2(j) <-> P_2(j) we can infer U_2(j) -> P_2(j). Finally, from U_2(j) -> P_2(j), r -> P_2(j), U_2(j) or r we can infer P_2(j), which contradicts ~P_2(j). Therefore, the answer is no.