Consider the following premises. exists x: ~T_2(x). forall x: U_1(x) or ~T_2(x). forall x: T_2(x) or ~U_1(x). Can we infer forall x: U_1(x) from them? If possible, name the inference rules being used at each step.
 From forall x: U_1(x) or ~T_2(x) we can infer forall x: T_2(x) -> U_1(x) via universal rewriting. From forall x: T_2(x) or ~U_1(x) we can infer forall x: U_1(x) -> T_2(x) via universal rewriting. From forall x: T_2(x) -> U_1(x), forall x: U_1(x) -> T_2(x) we can infer forall x: T_2(x) <-> U_1(x) via universal biconditional introduction. Finally, from forall x: T_2(x) <-> U_1(x), exists x: ~T_2(x) we can infer exists x: ~U_1(x) via existential biconditional elimination, which contradicts forall x: U_1(x). Therefore, the answer is no.