Consider the following premises. forall x_8: R_2(x_8) <-> Q_8(x_8). forall x_8: P_4(x_8) -> R_2(x_8). forall x_8: P_4(x_8) or Q_8(x_8). Can we infer exists x_8: ~(R_2(x_8) or W(x_8)) from them? If possible, name the inference rules being used at each step.
 From forall x_8: R_2(x_8) <-> Q_8(x_8) we can infer forall x_8: Q_8(x_8) -> R_2(x_8) via universal biconditional elimination. From forall x_8: P_4(x_8) -> R_2(x_8), forall x_8: Q_8(x_8) -> R_2(x_8), forall x_8: P_4(x_8) or Q_8(x_8) we can infer forall x_8: R_2(x_8) via universal disjunction elimination. Finally, from forall x_8: R_2(x_8) we can infer forall x_8: R_2(x_8) or W(x_8) via universal addition, which contradicts exists x_8: ~(R_2(x_8) or W(x_8)). Therefore, the answer is no.