Consider the following premises. forall x_14: Q_4(x_14). forall x_14: P_14(x_14) <-> Q_14(x_14). forall x_14: Q(x_14) -> R_4(x_14). forall x_14: Q_4(x_14) -> P_14(x_14). exists x_14: Q_14(x_14) -> R_4(x_14). Can we infer forall x_14: ~R_4(x_14) from them? If possible, name the inference rules being used at each step.
 From forall x_14: P_14(x_14) <-> Q_14(x_14) we can infer forall x_14: P_14(x_14) -> Q_14(x_14) via universal biconditional elimination. From forall x_14: Q_4(x_14) -> P_14(x_14), forall x_14: P_14(x_14) -> Q_14(x_14) we can infer forall x_14: Q_4(x_14) -> Q_14(x_14) via universal transitivity. From forall x_14: Q_4(x_14) -> Q_14(x_14), exists x_14: Q_14(x_14) -> R_4(x_14) we can infer exists x_14: Q_4(x_14) -> R_4(x_14) via existential transitivity. From forall x_14: Q_4(x_14) we can infer forall x_14: Q_4(x_14) or Q(x_14) via universal addition. Finally, from exists x_14: Q_4(x_14) -> R_4(x_14), forall x_14: Q(x_14) -> R_4(x_14), forall x_14: Q_4(x_14) or Q(x_14) we can infer exists x_14: R_4(x_14) via existential disjunction elimination, which contradicts forall x_14: ~R_4(x_14). Therefore, the answer is no.