Consider the following premises. forall x_7: Q_2(x_7) -> Q_7(x_7). exists x_7: Q_7(x_7) -> R_7(x_7). forall x_7: R_7(x_7) -> Q_2(x_7). forall x_7: Q_2(x_7) -> R_7(x_7). Can we infer exists x_7: Q_2(x_7) <-> R_7(x_7) from them? If possible, name the inference rules being used at each step.
 From forall x_7: Q_2(x_7) -> Q_7(x_7), exists x_7: Q_7(x_7) -> R_7(x_7) we can infer exists x_7: Q_2(x_7) -> R_7(x_7) via existential transitivity. From forall x_7: R_7(x_7) -> Q_2(x_7), forall x_7: Q_2(x_7) -> R_7(x_7) we can infer forall x_7: R_7(x_7) <-> Q_2(x_7) via universal biconditional introduction. From forall x_7: R_7(x_7) <-> Q_2(x_7) we can infer forall x_7: R_7(x_7) -> Q_2(x_7) via universal biconditional elimination. Finally, from exists x_7: Q_2(x_7) -> R_7(x_7), forall x_7: R_7(x_7) -> Q_2(x_7) we can infer exists x_7: Q_2(x_7) <-> R_7(x_7) via existential biconditional introduction. Therefore, the answer is yes.