Consider the following premises. forall x_4: Q_4(x_4) -> R_4(x_4). forall x_4: P_4(x_4) or Q_4(x_4). forall x_4: P_4(x_4) -> R_4(x_4). Can we infer exists x_4: ~(R_4(x_4) or Q(x_4)) from them? If possible, name the inference rules being used at each step.
 From forall x_4: P_4(x_4) -> R_4(x_4), forall x_4: Q_4(x_4) -> R_4(x_4), forall x_4: P_4(x_4) or Q_4(x_4) we can infer forall x_4: R_4(x_4) via universal disjunction elimination. Finally, from forall x_4: R_4(x_4) we can infer forall x_4: R_4(x_4) or Q(x_4) via universal addition, which contradicts exists x_4: ~(R_4(x_4) or Q(x_4)). Therefore, the answer is no.