Consider the following premises. forall x_18: P_18(x_18) <-> Q_18(x_18). forall x_18: Q_13(x_18) -> Q_18(x_18). forall x_18: ~P_18(x_18). exists x_18: Q_18(x_18) -> Q_13(x_18). forall x_18: T_2(x_18) <-> Q_13(x_18). Can we infer forall x_18: T_2(x_18) from them? If possible, name the inference rules being used at each step.
 From forall x_18: P_18(x_18) <-> Q_18(x_18) we can infer forall x_18: Q_18(x_18) -> P_18(x_18) via universal biconditional elimination. From exists x_18: Q_18(x_18) -> Q_13(x_18), forall x_18: Q_13(x_18) -> Q_18(x_18) we can infer exists x_18: Q_18(x_18) <-> Q_13(x_18) via existential biconditional introduction. From forall x_18: Q_18(x_18) -> P_18(x_18), forall x_18: ~P_18(x_18) we can infer forall x_18: ~Q_18(x_18) via universal modus tollens. From exists x_18: Q_18(x_18) <-> Q_13(x_18), forall x_18: ~Q_18(x_18) we can infer exists x_18: ~Q_13(x_18) via existential biconditional elimination. Finally, from forall x_18: T_2(x_18) <-> Q_13(x_18), exists x_18: ~Q_13(x_18) we can infer exists x_18: ~T_2(x_18) via existential biconditional elimination, which contradicts forall x_18: T_2(x_18). Therefore, the answer is no.