Consider the following premises. forall x_10: Q_5(x_10) -> P_4(x_10). forall x_10: P_4(x_10) -> P_10(x_10). forall x_10: P_10(x_10) -> P_4(x_10). exists x_10: P_10(x_10) -> Q_5(x_10). Can we infer exists x_10: P_4(x_10) <-> P_10(x_10) from them? If possible, name the inference rules being used at each step.
 From forall x_10: P_10(x_10) -> P_4(x_10), forall x_10: P_4(x_10) -> P_10(x_10) we can infer forall x_10: P_10(x_10) <-> P_4(x_10) via universal biconditional introduction. From exists x_10: P_10(x_10) -> Q_5(x_10), forall x_10: Q_5(x_10) -> P_4(x_10) we can infer exists x_10: P_10(x_10) -> P_4(x_10) via existential transitivity. From forall x_10: P_10(x_10) <-> P_4(x_10) we can infer forall x_10: P_4(x_10) -> P_10(x_10) via universal biconditional elimination. Finally, from forall x_10: P_4(x_10) -> P_10(x_10), exists x_10: P_10(x_10) -> P_4(x_10) we can infer exists x_10: P_4(x_10) <-> P_10(x_10) via existential biconditional introduction. Therefore, the answer is yes.