Consider the following premises. forall x_7: Q_7(x_7) -> P_4(x_7). forall x_7: P_6(x_7) or P_7(x_7). forall x_7: P_7(x_7) -> Q_7(x_7). forall x_7: P_6(x_7) -> P_4(x_7). forall x_7: ~P_4(x_7) or R(x_7). Can we infer forall x_7: Q_3(x_7) or R(x_7) from them? If possible, name the inference rules being used at each step.
 From forall x_7: P_7(x_7) -> Q_7(x_7), forall x_7: Q_7(x_7) -> P_4(x_7) we can infer forall x_7: P_7(x_7) -> P_4(x_7) via universal transitivity. From forall x_7: P_6(x_7) -> P_4(x_7), forall x_7: P_7(x_7) -> P_4(x_7), forall x_7: P_6(x_7) or P_7(x_7) we can infer forall x_7: P_4(x_7) via universal disjunction elimination. From forall x_7: P_4(x_7) we can infer forall x_7: P_4(x_7) or Q_3(x_7) via universal addition. Finally, from forall x_7: P_4(x_7) or Q_3(x_7), forall x_7: ~P_4(x_7) or R(x_7) we can infer forall x_7: Q_3(x_7) or R(x_7) via universal resolution. Therefore, the answer is yes.