Definition bool_fun_independent (bf : bool_fun) (x : BDDvar) :=
  forall vb : var_binding,
  bool_fun_eval bf (augment vb x true) =
  bool_fun_eval bf (augment vb x false).

Lemma bool_fun_independent_lemma : forall (bf : bool_fun) (x : BDDvar) (vb : var_binding) (b : bool), bool_fun_ext bf -> bool_fun_independent bf x -> bool_fun_eval bf (augment vb x b) = bool_fun_eval bf vb.