Axiom
  dependent_unique_choice :
    forall (A:Type) (B:A -> Type) (R:forall x:A, B x -> Prop),
      (forall x : A, exists! y : B x, R x y) ->
      (exists f : (forall x:A, B x), forall x:A, R x (f x)).

Theorem classic_set_in_prop_context : forall C:Prop, ((forall P:Prop, {P} + {~ P}) -> C) -> C.