Consider the following premises. forall z_3: U(z_3) or ~P_2(z_3). exists z_3: ~P_2(z_3). forall z_3: U(z_3) -> P_2(z_3). Can we infer forall z_3: U(z_3) from them? If possible, name the inference rules being used at each step.
 From forall z_3: U(z_3) or ~P_2(z_3) we can infer forall z_3: P_2(z_3) -> U(z_3) via universal rewriting. From forall z_3: P_2(z_3) -> U(z_3), forall z_3: U(z_3) -> P_2(z_3) we can infer forall z_3: P_2(z_3) <-> U(z_3) via universal biconditional introduction. Finally, from forall z_3: P_2(z_3) <-> U(z_3), exists z_3: ~P_2(z_3) we can infer exists z_3: ~U(z_3) via existential biconditional elimination, which contradicts forall z_3: U(z_3). Therefore, the answer is no.