Consider the following premises. forall x_5: S(x_5) -> U_2(x_5). forall x_5: Q_5(x_5) -> T_1(x_5). exists x_5: T_1(x_5) -> S(x_5). forall x_5: U_2(x_5) -> Q_5(x_5). Can we infer exists x_5: S(x_5) <-> T_1(x_5) from them? If possible, name the inference rules being used at each step.
 From forall x_5: U_2(x_5) -> Q_5(x_5), forall x_5: Q_5(x_5) -> T_1(x_5) we can infer forall x_5: U_2(x_5) -> T_1(x_5) via universal transitivity. From forall x_5: S(x_5) -> U_2(x_5), forall x_5: U_2(x_5) -> T_1(x_5) we can infer forall x_5: S(x_5) -> T_1(x_5) via universal transitivity. Finally, from forall x_5: S(x_5) -> T_1(x_5), exists x_5: T_1(x_5) -> S(x_5) we can infer exists x_5: S(x_5) <-> T_1(x_5) via existential biconditional introduction. Therefore, the answer is yes.