Consider the following premises. exists x_7: Q_7(x_7) -> P(x_7). forall x_7: Q_7(x_7) -> P(x_7). forall x_7: P(x_7) -> Q_7(x_7). Can we infer forall x_7: ~(P(x_7) <-> Q_7(x_7)) from them? If possible, name the inference rules being used at each step.
 From forall x_7: P(x_7) -> Q_7(x_7), exists x_7: Q_7(x_7) -> P(x_7) we can infer exists x_7: P(x_7) <-> Q_7(x_7) via existential biconditional introduction. From exists x_7: P(x_7) <-> Q_7(x_7) we can infer exists x_7: P(x_7) -> Q_7(x_7) via existential biconditional elimination. Finally, from exists x_7: P(x_7) -> Q_7(x_7), forall x_7: Q_7(x_7) -> P(x_7) we can infer exists x_7: P(x_7) <-> Q_7(x_7) via existential biconditional introduction, which contradicts forall x_7: ~(P(x_7) <-> Q_7(x_7)). Therefore, the answer is no.