Consider the following premises. forall x_10: T_2(x_10) -> Q_10(x_10). forall x_10: Q_10(x_10) -> T_2(x_10). forall x_10: T(x_10) -> R_1(x_10). forall x_10: R_1(x_10) -> T(x_10). forall x_10: ~Q_10(x_10). forall x_10: T(x_10) <-> T_2(x_10). Can we infer exists x_10: R_1(x_10) from them? If possible, name the inference rules being used at each step.
 From forall x_10: T_2(x_10) -> Q_10(x_10), forall x_10: Q_10(x_10) -> T_2(x_10) we can infer forall x_10: T_2(x_10) <-> Q_10(x_10) via universal biconditional introduction. From forall x_10: T(x_10) -> R_1(x_10), forall x_10: R_1(x_10) -> T(x_10) we can infer forall x_10: T(x_10) <-> R_1(x_10) via universal biconditional introduction. From forall x_10: T_2(x_10) <-> Q_10(x_10), forall x_10: ~Q_10(x_10) we can infer forall x_10: ~T_2(x_10) via universal biconditional elimination. From forall x_10: T(x_10) <-> T_2(x_10), forall x_10: ~T_2(x_10) we can infer forall x_10: ~T(x_10) via universal biconditional elimination. Finally, from forall x_10: T(x_10) <-> R_1(x_10), forall x_10: ~T(x_10) we can infer forall x_10: ~R_1(x_10) via universal biconditional elimination, which contradicts exists x_10: R_1(x_10). Therefore, the answer is no.