Consider the following premises. forall x_2: ~Q_2(x_2). forall x_2: P_2(x_2) -> Q_2(x_2). forall x_2: U(x_2) -> P_2(x_2). forall x_2: P_2(x_2) -> U(x_2). Can we infer forall x_2: ~U(x_2) from them? If possible, name the inference rules being used at each step.
 From forall x_2: P_2(x_2) -> Q_2(x_2), forall x_2: ~Q_2(x_2) we can infer forall x_2: ~P_2(x_2) via universal modus tollens. From forall x_2: U(x_2) -> P_2(x_2), forall x_2: P_2(x_2) -> U(x_2) we can infer forall x_2: U(x_2) <-> P_2(x_2) via universal biconditional introduction. Finally, from forall x_2: U(x_2) <-> P_2(x_2), forall x_2: ~P_2(x_2) we can infer forall x_2: ~U(x_2) via universal biconditional elimination. Therefore, the answer is yes.