Consider the following premises. forall x_14: P_14(x_14) or ~Q_11(x_14). forall x_14: W(x_14) -> S_2(x_14). forall x_14: Q_14(x_14) -> W(x_14). exists x_14: S_2(x_14) -> Q_11(x_14). forall x_14: P_14(x_14) -> Q_14(x_14). Can we infer forall x_14: ~(W(x_14) <-> S_2(x_14)) from them? If possible, name the inference rules being used at each step.
 From forall x_14: P_14(x_14) -> Q_14(x_14), forall x_14: Q_14(x_14) -> W(x_14) we can infer forall x_14: P_14(x_14) -> W(x_14) via universal transitivity. From forall x_14: P_14(x_14) or ~Q_11(x_14) we can infer forall x_14: Q_11(x_14) -> P_14(x_14) via universal rewriting. From forall x_14: Q_11(x_14) -> P_14(x_14), forall x_14: P_14(x_14) -> W(x_14) we can infer forall x_14: Q_11(x_14) -> W(x_14) via universal transitivity. From exists x_14: S_2(x_14) -> Q_11(x_14), forall x_14: Q_11(x_14) -> W(x_14) we can infer exists x_14: S_2(x_14) -> W(x_14) via existential transitivity. Finally, from forall x_14: W(x_14) -> S_2(x_14), exists x_14: S_2(x_14) -> W(x_14) we can infer exists x_14: W(x_14) <-> S_2(x_14) via existential biconditional introduction, which contradicts forall x_14: ~(W(x_14) <-> S_2(x_14)). Therefore, the answer is no.