Consider the following premises. exists x_11: S_4(x_11) -> Q_11(x_11). forall x_11: Q_11(x_11) -> S_4(x_11). forall x_11: ~Q_7(x_11). forall x_11: S_4(x_11) -> Q_11(x_11). forall x_11: S_4(x_11) -> Q_7(x_11). Can we infer exists x_11: ~Q_11(x_11) from them? If possible, name the inference rules being used at each step.
 From forall x_11: S_4(x_11) -> Q_11(x_11), forall x_11: Q_11(x_11) -> S_4(x_11) we can infer forall x_11: S_4(x_11) <-> Q_11(x_11) via universal biconditional introduction. From forall x_11: S_4(x_11) -> Q_7(x_11), forall x_11: ~Q_7(x_11) we can infer forall x_11: ~S_4(x_11) via universal modus tollens. From forall x_11: S_4(x_11) <-> Q_11(x_11) we can infer forall x_11: Q_11(x_11) -> S_4(x_11) via universal biconditional elimination. From exists x_11: S_4(x_11) -> Q_11(x_11), forall x_11: Q_11(x_11) -> S_4(x_11) we can infer exists x_11: S_4(x_11) <-> Q_11(x_11) via existential biconditional introduction. Finally, from exists x_11: S_4(x_11) <-> Q_11(x_11), forall x_11: ~S_4(x_11) we can infer exists x_11: ~Q_11(x_11) via existential biconditional elimination. Therefore, the answer is yes.