Consider the following premises. forall x_7: P_7(x_7) <-> Q_7(x_7). forall x_7: P_2(x_7) -> ~Q_7(x_7). forall x_7: P_7(x_7). Can we infer forall x_7: ~P_2(x_7) from them? If possible, name the inference rules being used at each step.
 From forall x_7: P_7(x_7) <-> Q_7(x_7) we can infer forall x_7: P_7(x_7) -> Q_7(x_7) via universal biconditional elimination. From forall x_7: P_7(x_7) -> Q_7(x_7), forall x_7: P_7(x_7) we can infer forall x_7: Q_7(x_7) via universal modus ponens. Finally, from forall x_7: P_2(x_7) -> ~Q_7(x_7), forall x_7: Q_7(x_7) we can infer forall x_7: ~P_2(x_7) via universal modus tollens. Therefore, the answer is yes.