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