Consider the following premises. exists x_10: P_1(x_10) -> Q_10(x_10). forall x_10: P_1(x_10) or ~Q_7(x_10). forall x_10: R_10(x_10) -> Q_7(x_10). forall x_10: Q_10(x_10) -> R_10(x_10). Can we infer exists x_10: P_1(x_10) <-> Q_7(x_10) from them? If possible, name the inference rules being used at each step.
 From exists x_10: P_1(x_10) -> Q_10(x_10), forall x_10: Q_10(x_10) -> R_10(x_10) we can infer exists x_10: P_1(x_10) -> R_10(x_10) via existential transitivity. From forall x_10: P_1(x_10) or ~Q_7(x_10) we can infer forall x_10: Q_7(x_10) -> P_1(x_10) via universal rewriting. From exists x_10: P_1(x_10) -> R_10(x_10), forall x_10: R_10(x_10) -> Q_7(x_10) we can infer exists x_10: P_1(x_10) -> Q_7(x_10) via existential transitivity. Finally, from exists x_10: P_1(x_10) -> Q_7(x_10), forall x_10: Q_7(x_10) -> P_1(x_10) we can infer exists x_10: P_1(x_10) <-> Q_7(x_10) via existential biconditional introduction. Therefore, the answer is yes.