Consider the following premises. forall x_7: Q_1(x_7) -> Q_7(x_7). forall x_7: S(x_7) -> Q_1(x_7). exists x_7: S(x_7) or ~Q_7(x_7). Can we infer forall x_7: ~(S(x_7) <-> Q_7(x_7)) from them? If possible, name the inference rules being used at each step.
 From exists x_7: S(x_7) or ~Q_7(x_7) we can infer exists x_7: Q_7(x_7) -> S(x_7) via existential rewriting. From forall x_7: S(x_7) -> Q_1(x_7), forall x_7: Q_1(x_7) -> Q_7(x_7) we can infer forall x_7: S(x_7) -> Q_7(x_7) via universal transitivity. Finally, from forall x_7: S(x_7) -> Q_7(x_7), exists x_7: Q_7(x_7) -> S(x_7) we can infer exists x_7: S(x_7) <-> Q_7(x_7) via existential biconditional introduction, which contradicts forall x_7: ~(S(x_7) <-> Q_7(x_7)). Therefore, the answer is no.