Consider the following premises. exists x_9: P_9(x_9) or ~P(x_9). forall x_9: P(x_9) or ~P_9(x_9). Can we infer exists x_9: P_9(x_9) -> P(x_9) from them? If possible, name the inference rules being used at each step.
 From exists x_9: P_9(x_9) or ~P(x_9) we can infer exists x_9: P(x_9) -> P_9(x_9) via existential rewriting. From forall x_9: P(x_9) or ~P_9(x_9) we can infer forall x_9: P_9(x_9) -> P(x_9) via universal rewriting. From exists x_9: P(x_9) -> P_9(x_9), forall x_9: P_9(x_9) -> P(x_9) we can infer exists x_9: P(x_9) <-> P_9(x_9) via existential biconditional introduction. Finally, from exists x_9: P(x_9) <-> P_9(x_9) we can infer exists x_9: P_9(x_9) -> P(x_9) via existential biconditional elimination. Therefore, the answer is yes.