Consider the following premises. forall y_4: ~P(y_4). forall y_4: P(y_4) -> U(y_4). forall y_4: U(y_4) -> P(y_4). forall y_4: U(y_4) or P_3(y_4). Can we infer forall y_4: P_3(y_4) from them? If possible, name the inference rules being used at each step.
 From forall y_4: U(y_4) -> P(y_4), forall y_4: P(y_4) -> U(y_4) we can infer forall y_4: U(y_4) <-> P(y_4) via universal biconditional introduction. From forall y_4: U(y_4) <-> P(y_4), forall y_4: ~P(y_4) we can infer forall y_4: ~U(y_4) via universal biconditional elimination. Finally, from forall y_4: U(y_4) or P_3(y_4), forall y_4: ~U(y_4) we can infer forall y_4: P_3(y_4) via universal disjunctive syllogism. Therefore, the answer is yes.