Definition all_equiv_under (l1 l2 : list Prop) :=
forall x y z, In x l1 -> In y l2 -> In z l2 -> (x -> (y <-> z)).

Lemma all_equiv2_impl2__all_equiv : forall l1 l2 x y x' y',
all_equiv l1 -> all_equiv l2 ->
In x l1 -> In y l2 -> (x -> y) ->
In x' l2 -> In y' l1 -> (x' -> y') ->
all_equiv (l1++l2).