Consider the following premises. forall x_11: P_7(x_11) or Q_7(x_11). forall x_11: ~P_7(x_11). forall x_11: R_11(x_11) -> R_5(x_11). forall x_11: Q_11(x_11) -> R_5(x_11). forall x_11: ~P_11(x_11) or R_11(x_11). forall x_11: P_11(x_11) or Q_11(x_11). Can we infer exists x_11: ~Q_7(x_11) from them? If possible, name the inference rules being used at each step.
 From forall x_11: P_11(x_11) or Q_11(x_11), forall x_11: ~P_11(x_11) or R_11(x_11) we can infer forall x_11: Q_11(x_11) or R_11(x_11) via universal resolution. From forall x_11: P_7(x_11) or Q_7(x_11), forall x_11: ~P_7(x_11) we can infer forall x_11: Q_7(x_11) via universal disjunctive syllogism. From forall x_11: Q_11(x_11) -> R_5(x_11), forall x_11: R_11(x_11) -> R_5(x_11), forall x_11: Q_11(x_11) or R_11(x_11) we can infer forall x_11: R_5(x_11) via universal disjunction elimination. From forall x_11: R_5(x_11), forall x_11: Q_7(x_11) we can infer forall x_11: R_5(x_11) and Q_7(x_11) via universal conjunction. Finally, from forall x_11: R_5(x_11) and Q_7(x_11) we can infer forall x_11: R_5(x_11) via universal simplification, which contradicts exists x_11: ~Q_7(x_11). Therefore, the answer is no.