Consider the following premises. forall x_12: ~U(x_12). forall x_12: P_5(x_12) -> Q_12(x_12). forall x_12: P(x_12) -> P_5(x_12). forall x_12: U(x_12) <-> Q_12(x_12). Can we infer forall x_12: ~P(x_12) from them? If possible, name the inference rules being used at each step.
 From forall x_12: U(x_12) <-> Q_12(x_12) we can infer forall x_12: Q_12(x_12) -> U(x_12) via universal biconditional elimination. From forall x_12: P_5(x_12) -> Q_12(x_12), forall x_12: Q_12(x_12) -> U(x_12) we can infer forall x_12: P_5(x_12) -> U(x_12) via universal transitivity. From forall x_12: P(x_12) -> P_5(x_12), forall x_12: P_5(x_12) -> U(x_12) we can infer forall x_12: P(x_12) -> U(x_12) via universal transitivity. Finally, from forall x_12: P(x_12) -> U(x_12), forall x_12: ~U(x_12) we can infer forall x_12: ~P(x_12) via universal modus tollens. Therefore, the answer is yes.