Consider the following premises. forall x_9: Q_1(x_9) -> S_3(x_9). forall x_9: W_2(x_9) -> Q_1(x_9). exists x_9: W_2(x_9) <-> S_3(x_9). Can we infer forall x_9: ~(W_2(x_9) <-> S_3(x_9)) from them? If possible, name the inference rules being used at each step.
 From exists x_9: W_2(x_9) <-> S_3(x_9) we can infer exists x_9: S_3(x_9) -> W_2(x_9) via existential biconditional elimination. From forall x_9: W_2(x_9) -> Q_1(x_9), forall x_9: Q_1(x_9) -> S_3(x_9) we can infer forall x_9: W_2(x_9) -> S_3(x_9) via universal transitivity. Finally, from forall x_9: W_2(x_9) -> S_3(x_9), exists x_9: S_3(x_9) -> W_2(x_9) we can infer exists x_9: W_2(x_9) <-> S_3(x_9) via existential biconditional introduction, which contradicts forall x_9: ~(W_2(x_9) <-> S_3(x_9)). Therefore, the answer is no.