Consider the following premises. forall x_7: P_7(x_7) or ~Q_7(x_7). exists x_7: Q_7(x_7) <-> Q_5(x_7). Can we infer exists x_7: Q_5(x_7) -> P_7(x_7) from them? If possible, name the inference rules being used at each step.
 From forall x_7: P_7(x_7) or ~Q_7(x_7) we can infer forall x_7: Q_7(x_7) -> P_7(x_7) via universal rewriting. From exists x_7: Q_7(x_7) <-> Q_5(x_7) we can infer exists x_7: Q_5(x_7) -> Q_7(x_7) via existential biconditional elimination. Finally, from exists x_7: Q_5(x_7) -> Q_7(x_7), forall x_7: Q_7(x_7) -> P_7(x_7) we can infer exists x_7: Q_5(x_7) -> P_7(x_7) via existential transitivity. Therefore, the answer is yes.