Consider the following premises. forall x_12: T_4(x_12) <-> U(x_12). exists x_12: ~Q_12(x_12). forall x_12: T_4(x_12) <-> Q_12(x_12). Can we infer exists x_12: ~U(x_12) from them? If possible, name the inference rules being used at each step.
 From forall x_12: T_4(x_12) <-> Q_12(x_12), exists x_12: ~Q_12(x_12) we can infer exists x_12: ~T_4(x_12) via existential biconditional elimination. From forall x_12: T_4(x_12) <-> U(x_12) we can infer forall x_12: U(x_12) -> T_4(x_12) via universal biconditional elimination. Finally, from forall x_12: U(x_12) -> T_4(x_12), exists x_12: ~T_4(x_12) we can infer exists x_12: ~U(x_12) via existential modus tollens. Therefore, the answer is yes.