Consider the following premises. forall x_17: P_7(x_17) <-> Q_7(x_17). forall x_17: P_17(x_17) -> R_17(x_17). forall x_17: P_4(x_17) or P_7(x_17). forall x_17: P_4(x_17) -> Q_7(x_17). forall x_17: Q_17(x_17) -> R_17(x_17). exists x_17: P_17(x_17) or Q_17(x_17). Can we infer forall x_17: ~(R_17(x_17) and Q_7(x_17)) from them? If possible, name the inference rules being used at each step.
 From forall x_17: P_17(x_17) -> R_17(x_17), forall x_17: Q_17(x_17) -> R_17(x_17), exists x_17: P_17(x_17) or Q_17(x_17) we can infer exists x_17: R_17(x_17) via existential disjunction elimination. From forall x_17: P_7(x_17) <-> Q_7(x_17) we can infer forall x_17: P_7(x_17) -> Q_7(x_17) via universal biconditional elimination. From forall x_17: P_4(x_17) -> Q_7(x_17), forall x_17: P_7(x_17) -> Q_7(x_17), forall x_17: P_4(x_17) or P_7(x_17) we can infer forall x_17: Q_7(x_17) via universal disjunction elimination. Finally, from exists x_17: R_17(x_17), forall x_17: Q_7(x_17) we can infer exists x_17: R_17(x_17) and Q_7(x_17) via existential conjunction, which contradicts forall x_17: ~(R_17(x_17) and Q_7(x_17)). Therefore, the answer is no.