Consider the following premises. exists x_13: Q_9(x_13) -> T(x_13). forall x_13: ~R_13(x_13). forall x_13: T(x_13) -> Q_13(x_13). forall x_13: Q_13(x_13) -> R_13(x_13). forall x_13: R_13(x_13) -> Q_9(x_13). Can we infer exists x_13: ~T(x_13) from them? If possible, name the inference rules being used at each step.
 From forall x_13: T(x_13) -> Q_13(x_13), forall x_13: Q_13(x_13) -> R_13(x_13) we can infer forall x_13: T(x_13) -> R_13(x_13) via universal transitivity. From forall x_13: R_13(x_13) -> Q_9(x_13), exists x_13: Q_9(x_13) -> T(x_13) we can infer exists x_13: R_13(x_13) -> T(x_13) via existential transitivity. From forall x_13: T(x_13) -> R_13(x_13), exists x_13: R_13(x_13) -> T(x_13) we can infer exists x_13: T(x_13) <-> R_13(x_13) via existential biconditional introduction. From exists x_13: T(x_13) <-> R_13(x_13) we can infer exists x_13: T(x_13) -> R_13(x_13) via existential biconditional elimination. Finally, from exists x_13: T(x_13) -> R_13(x_13), forall x_13: ~R_13(x_13) we can infer exists x_13: ~T(x_13) via existential modus tollens. Therefore, the answer is yes.