Consider the following premises. exists x_7: S_3(x_7) -> P_7(x_7). forall x_7: P_7(x_7) -> S_3(x_7). forall x_7: R(x_7) -> S_3(x_7). Can we infer exists x_7: R(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) -> S_3(x_7), exists x_7: S_3(x_7) -> P_7(x_7) we can infer exists x_7: P_7(x_7) <-> S_3(x_7) via existential biconditional introduction. From exists x_7: P_7(x_7) <-> S_3(x_7) we can infer exists x_7: S_3(x_7) -> P_7(x_7) via existential biconditional elimination. Finally, from forall x_7: R(x_7) -> S_3(x_7), exists x_7: S_3(x_7) -> P_7(x_7) we can infer exists x_7: R(x_7) -> P_7(x_7) via existential transitivity. Therefore, the answer is yes.