Consider the following premises. forall x_12: Q_9(x_12) -> W(x_12). exists x_12: U_4(x_12) or ~W(x_12). forall x_12: U_4(x_12) -> Q_9(x_12). forall x_12: Q_9(x_12) -> U_4(x_12). Can we infer forall x_12: ~(W(x_12) <-> U_4(x_12)) from them? If possible, name the inference rules being used at each step.
 From exists x_12: U_4(x_12) or ~W(x_12) we can infer exists x_12: W(x_12) -> U_4(x_12) via existential rewriting. Finally, from forall x_12: U_4(x_12) -> Q_9(x_12), forall x_12: Q_9(x_12) -> U_4(x_12) we can infer forall x_12: U_4(x_12) <-> Q_9(x_12) via universal biconditional introduction, which contradicts forall x_12: ~(W(x_12) <-> U_4(x_12)). Therefore, the answer is no.