Consider the following premises. exists x_6: ~S(x_6). forall x_6: Q_6(x_6) -> S_4(x_6). forall x_6: S(x_6) or ~S_4(x_6). forall x_6: S(x_6) -> Q_6(x_6). Can we infer exists x_6: ~S_4(x_6) from them? If possible, name the inference rules being used at each step.
 From forall x_6: S(x_6) -> Q_6(x_6), forall x_6: Q_6(x_6) -> S_4(x_6) we can infer forall x_6: S(x_6) -> S_4(x_6) via universal transitivity. From forall x_6: S(x_6) or ~S_4(x_6) we can infer forall x_6: S_4(x_6) -> S(x_6) via universal rewriting. From forall x_6: S_4(x_6) -> S(x_6), forall x_6: S(x_6) -> S_4(x_6) we can infer forall x_6: S_4(x_6) <-> S(x_6) via universal biconditional introduction. Finally, from forall x_6: S_4(x_6) <-> S(x_6), exists x_6: ~S(x_6) we can infer exists x_6: ~S_4(x_6) via existential biconditional elimination. Therefore, the answer is yes.