Consider the following premises. exists x_3: S_3(x_3) -> W(x_3). forall x_3: ~Q_3(x_3). forall x_3: W(x_3) -> S_3(x_3). forall x_3: W(x_3) -> Q_3(x_3). Can we infer exists x_3: ~S_3(x_3) from them? If possible, name the inference rules being used at each step.
 From forall x_3: W(x_3) -> Q_3(x_3), forall x_3: ~Q_3(x_3) we can infer forall x_3: ~W(x_3) via universal modus tollens. From exists x_3: S_3(x_3) -> W(x_3), forall x_3: W(x_3) -> S_3(x_3) we can infer exists x_3: S_3(x_3) <-> W(x_3) via existential biconditional introduction. Finally, from exists x_3: S_3(x_3) <-> W(x_3), forall x_3: ~W(x_3) we can infer exists x_3: ~S_3(x_3) via existential biconditional elimination. Therefore, the answer is yes.