Consider the following premises. forall y_4: P(y_4) or ~R_3(y_4). exists y_4: R_3(y_4) or ~P(y_4). Can we infer exists y_4: R_3(y_4) -> P(y_4) from them? If possible, name the inference rules being used at each step.
 From exists y_4: R_3(y_4) or ~P(y_4) we can infer exists y_4: P(y_4) -> R_3(y_4) via existential rewriting. From forall y_4: P(y_4) or ~R_3(y_4) we can infer forall y_4: R_3(y_4) -> P(y_4) via universal rewriting. From exists y_4: P(y_4) -> R_3(y_4), forall y_4: R_3(y_4) -> P(y_4) we can infer exists y_4: P(y_4) <-> R_3(y_4) via existential biconditional introduction. Finally, from exists y_4: P(y_4) <-> R_3(y_4) we can infer exists y_4: R_3(y_4) -> P(y_4) via existential biconditional elimination. Therefore, the answer is yes.