Consider the following premises. forall x_11: R_2(x_11) -> Q_11(x_11). forall x_11: P_4(x_11) or ~Q_11(x_11). exists x_11: R(x_11) <-> P_4(x_11). forall x_11: R_2(x_11) or R(x_11). Can we infer forall x_11: ~P_4(x_11) from them? If possible, name the inference rules being used at each step.
 From forall x_11: P_4(x_11) or ~Q_11(x_11) we can infer forall x_11: Q_11(x_11) -> P_4(x_11) via universal rewriting. From exists x_11: R(x_11) <-> P_4(x_11) we can infer exists x_11: R(x_11) -> P_4(x_11) via existential biconditional elimination. From forall x_11: R_2(x_11) -> Q_11(x_11), forall x_11: Q_11(x_11) -> P_4(x_11) we can infer forall x_11: R_2(x_11) -> P_4(x_11) via universal transitivity. Finally, from forall x_11: R_2(x_11) -> P_4(x_11), exists x_11: R(x_11) -> P_4(x_11), forall x_11: R_2(x_11) or R(x_11) we can infer exists x_11: P_4(x_11) via existential disjunction elimination, which contradicts forall x_11: ~P_4(x_11). Therefore, the answer is no.