Consider the following premises. forall x_7: Q(x_7). forall x_7: Q_7(x_7) -> R_7(x_7). forall x_7: P_7(x_7) or Q_7(x_7). exists x_7: P_7(x_7) -> R_7(x_7). Can we infer forall x_7: ~(R_7(x_7) and Q(x_7)) from them? If possible, name the inference rules being used at each step.
 From exists x_7: P_7(x_7) -> R_7(x_7), forall x_7: Q_7(x_7) -> R_7(x_7), forall x_7: P_7(x_7) or Q_7(x_7) we can infer exists x_7: R_7(x_7) via existential disjunction elimination. Finally, from exists x_7: R_7(x_7), forall x_7: Q(x_7) we can infer exists x_7: R_7(x_7) and Q(x_7) via existential conjunction, which contradicts forall x_7: ~(R_7(x_7) and Q(x_7)). Therefore, the answer is no.