Consider the following premises. exists x_8: P(x_8) or ~Q_8(x_8). forall x_8: P(x_8) -> P_2(x_8). forall x_8: P_2(x_8) -> Q_8(x_8). Can we infer forall x_8: ~(P(x_8) <-> P_2(x_8)) from them? If possible, name the inference rules being used at each step.
 From exists x_8: P(x_8) or ~Q_8(x_8) we can infer exists x_8: Q_8(x_8) -> P(x_8) via existential rewriting. From forall x_8: P_2(x_8) -> Q_8(x_8), exists x_8: Q_8(x_8) -> P(x_8) we can infer exists x_8: P_2(x_8) -> P(x_8) via existential transitivity. Finally, from forall x_8: P(x_8) -> P_2(x_8), exists x_8: P_2(x_8) -> P(x_8) we can infer exists x_8: P(x_8) <-> P_2(x_8) via existential biconditional introduction, which contradicts forall x_8: ~(P(x_8) <-> P_2(x_8)). Therefore, the answer is no.