Consider the following premises. forall x_5: ~Q_5(x_5) or S(x_5). forall x_5: ~Q(x_5) or R(x_5). forall x_5: P_5(x_5). forall x_5: ~P_5(x_5) or Q(x_5). Can we infer exists x_5: ~(S(x_5) or R(x_5)) from them? If possible, name the inference rules being used at each step.
 From forall x_5: P_5(x_5) we can infer forall x_5: P_5(x_5) or Q_5(x_5) via universal addition. From forall x_5: P_5(x_5) or Q_5(x_5), forall x_5: ~P_5(x_5) or Q(x_5) we can infer forall x_5: Q_5(x_5) or Q(x_5) via universal resolution. From forall x_5: Q_5(x_5) or Q(x_5), forall x_5: ~Q_5(x_5) or S(x_5) we can infer forall x_5: Q(x_5) or S(x_5) via universal resolution. Finally, from forall x_5: Q(x_5) or S(x_5), forall x_5: ~Q(x_5) or R(x_5) we can infer forall x_5: S(x_5) or R(x_5) via universal resolution, which contradicts exists x_5: ~(S(x_5) or R(x_5)). Therefore, the answer is no.