Consider the following premises. forall x_4: S_4(x_4) <-> S(x_4). exists x_4: ~S_4(x_4). Can we infer exists x_4: ~S(x_4) from them? If possible, name the inference rules being used at each step.
 From forall x_4: S_4(x_4) <-> S(x_4) we can infer forall x_4: S(x_4) -> S_4(x_4) via universal biconditional elimination. Finally, from forall x_4: S(x_4) -> S_4(x_4), exists x_4: ~S_4(x_4) we can infer exists x_4: ~S(x_4) via existential modus tollens. Therefore, the answer is yes.