Consider the following premises. exists x_7: S_4(x_7). forall x_7: S_4(x_7) -> Q_7(x_7). forall x_7: S_4(x_7) or ~Q_7(x_7). Can we infer forall x_7: ~Q_7(x_7) from them? If possible, name the inference rules being used at each step.
 From forall x_7: S_4(x_7) or ~Q_7(x_7) we can infer forall x_7: Q_7(x_7) -> S_4(x_7) via universal rewriting. From forall x_7: S_4(x_7) -> Q_7(x_7), forall x_7: Q_7(x_7) -> S_4(x_7) we can infer forall x_7: S_4(x_7) <-> Q_7(x_7) via universal biconditional introduction. From forall x_7: S_4(x_7) <-> Q_7(x_7) we can infer forall x_7: S_4(x_7) -> Q_7(x_7) via universal biconditional elimination. Finally, from forall x_7: S_4(x_7) -> Q_7(x_7), exists x_7: S_4(x_7) we can infer exists x_7: Q_7(x_7) via existential modus ponens, which contradicts forall x_7: ~Q_7(x_7). Therefore, the answer is no.