Consider the following premises. forall x_3: P_4(x_3) -> U(x_3). exists x_3: P_3(x_3) or ~U(x_3). Can we infer exists x_3: P_4(x_3) -> P_3(x_3) from them? If possible, name the inference rules being used at each step.
 From exists x_3: P_3(x_3) or ~U(x_3) we can infer exists x_3: U(x_3) -> P_3(x_3) via existential rewriting. Finally, from forall x_3: P_4(x_3) -> U(x_3), exists x_3: U(x_3) -> P_3(x_3) we can infer exists x_3: P_4(x_3) -> P_3(x_3) via existential transitivity. Therefore, the answer is yes.