Consider the following premises. forall x: W_4(x) -> P(x). forall x: P(x) -> T(x). forall x: W_4(x) or ~T(x). Can we infer exists x: ~(W_4(x) <-> T(x)) from them? If possible, name the inference rules being used at each step.
 From forall x: W_4(x) -> P(x), forall x: P(x) -> T(x) we can infer forall x: W_4(x) -> T(x) via universal transitivity. From forall x: W_4(x) or ~T(x) we can infer forall x: T(x) -> W_4(x) via universal rewriting. Finally, from forall x: W_4(x) -> T(x), forall x: T(x) -> W_4(x) we can infer forall x: W_4(x) <-> T(x) via universal biconditional introduction, which contradicts exists x: ~(W_4(x) <-> T(x)). Therefore, the answer is no.