Consider the following premises. forall x_12: P_7(x_12) -> P_12(x_12). exists x_12: P_12(x_12) -> P_7(x_12). forall x_12: Q_12(x_12) -> P_12(x_12). forall x_12: P_12(x_12) -> Q_12(x_12). Can we infer exists x_12: P_7(x_12) -> Q_12(x_12) from them? If possible, name the inference rules being used at each step.
 From forall x_12: P_12(x_12) -> Q_12(x_12), forall x_12: Q_12(x_12) -> P_12(x_12) we can infer forall x_12: P_12(x_12) <-> Q_12(x_12) via universal biconditional introduction. From forall x_12: P_12(x_12) <-> Q_12(x_12) we can infer forall x_12: P_12(x_12) -> Q_12(x_12) via universal biconditional elimination. From forall x_12: P_7(x_12) -> P_12(x_12), exists x_12: P_12(x_12) -> P_7(x_12) we can infer exists x_12: P_7(x_12) <-> P_12(x_12) via existential biconditional introduction. From exists x_12: P_7(x_12) <-> P_12(x_12) we can infer exists x_12: P_7(x_12) -> P_12(x_12) via existential biconditional elimination. Finally, from exists x_12: P_7(x_12) -> P_12(x_12), forall x_12: P_12(x_12) -> Q_12(x_12) we can infer exists x_12: P_7(x_12) -> Q_12(x_12) via existential transitivity. Therefore, the answer is yes.