Consider the following premises. forall x_2: ~T(x_2). forall x_2: W(x_2) -> T(x_2). exists x_2: U(x_2) -> W(x_2). forall x_2: T(x_2) -> W(x_2). Can we infer exists x_2: ~U(x_2) from them? If possible, name the inference rules being used at each step.
 From forall x_2: W(x_2) -> T(x_2), forall x_2: T(x_2) -> W(x_2) we can infer forall x_2: W(x_2) <-> T(x_2) via universal biconditional introduction. From forall x_2: W(x_2) <-> T(x_2), forall x_2: ~T(x_2) we can infer forall x_2: ~W(x_2) via universal biconditional elimination. Finally, from exists x_2: U(x_2) -> W(x_2), forall x_2: ~W(x_2) we can infer exists x_2: ~U(x_2) via existential modus tollens. Therefore, the answer is yes.