Consider the following premises. forall x_6: U(x_6) or ~P(x_6). forall x_6: P_4(x_6) <-> U(x_6). Can we infer forall x_6: P(x_6) -> P_4(x_6) from them? If possible, name the inference rules being used at each step.
 From forall x_6: U(x_6) or ~P(x_6) we can infer forall x_6: P(x_6) -> U(x_6) via universal rewriting. From forall x_6: P_4(x_6) <-> U(x_6) we can infer forall x_6: U(x_6) -> P_4(x_6) via universal biconditional elimination. Finally, from forall x_6: P(x_6) -> U(x_6), forall x_6: U(x_6) -> P_4(x_6) we can infer forall x_6: P(x_6) -> P_4(x_6) via universal transitivity. Therefore, the answer is yes.