Consider the following premises. forall x_8: Q(x_8) -> Q_8(x_8). exists x_8: Q_8(x_8) -> P_4(x_8). forall x_8: R_6(x_8) -> Q(x_8). forall x_8: P_6(x_8) or Q_6(x_8). forall x_8: ~P_6(x_8) or R_6(x_8). forall x_8: Q_6(x_8) -> Q(x_8). Can we infer forall x_8: ~P_4(x_8) from them? If possible, name the inference rules being used at each step.
 From forall x_8: Q(x_8) -> Q_8(x_8), exists x_8: Q_8(x_8) -> P_4(x_8) we can infer exists x_8: Q(x_8) -> P_4(x_8) via existential transitivity. From forall x_8: P_6(x_8) or Q_6(x_8), forall x_8: ~P_6(x_8) or R_6(x_8) we can infer forall x_8: Q_6(x_8) or R_6(x_8) via universal resolution. From forall x_8: Q_6(x_8) -> Q(x_8), forall x_8: R_6(x_8) -> Q(x_8), forall x_8: Q_6(x_8) or R_6(x_8) we can infer forall x_8: Q(x_8) via universal disjunction elimination. Finally, from exists x_8: Q(x_8) -> P_4(x_8), forall x_8: Q(x_8) we can infer exists x_8: P_4(x_8) via existential modus ponens, which contradicts forall x_8: ~P_4(x_8). Therefore, the answer is no.