Consider the following premises. forall x_21: Q(x_21) -> W(x_21). forall x_21: S(x_21) -> Q_21(x_21). exists x_21: W_2(x_21). forall x_21: Q(x_21) <-> Q_21(x_21). forall x_21: W_2(x_21) -> S(x_21). Can we infer forall x_21: ~W(x_21) from them? If possible, name the inference rules being used at each step.
 From forall x_21: Q(x_21) <-> Q_21(x_21) we can infer forall x_21: Q_21(x_21) -> Q(x_21) via universal biconditional elimination. From forall x_21: S(x_21) -> Q_21(x_21), forall x_21: Q_21(x_21) -> Q(x_21) we can infer forall x_21: S(x_21) -> Q(x_21) via universal transitivity. From forall x_21: W_2(x_21) -> S(x_21), exists x_21: W_2(x_21) we can infer exists x_21: S(x_21) via existential modus ponens. From forall x_21: S(x_21) -> Q(x_21), forall x_21: Q(x_21) -> W(x_21) we can infer forall x_21: S(x_21) -> W(x_21) via universal transitivity. Finally, from forall x_21: S(x_21) -> W(x_21), exists x_21: S(x_21) we can infer exists x_21: W(x_21) via existential modus ponens, which contradicts forall x_21: ~W(x_21). Therefore, the answer is no.