Consider the following premises. exists x_4: ~T_1(x_4). forall x_4: T(x_4) <-> T_1(x_4). forall x_4: T(x_4) <-> U_2(x_4). Can we infer exists x_4: ~U_2(x_4) from them? If possible, name the inference rules being used at each step.
 From forall x_4: T(x_4) <-> U_2(x_4) we can infer forall x_4: U_2(x_4) -> T(x_4) via universal biconditional elimination. From forall x_4: T(x_4) <-> T_1(x_4), exists x_4: ~T_1(x_4) we can infer exists x_4: ~T(x_4) via existential biconditional elimination. Finally, from forall x_4: U_2(x_4) -> T(x_4), exists x_4: ~T(x_4) we can infer exists x_4: ~U_2(x_4) via existential modus tollens. Therefore, the answer is yes.