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_equiv_under_chara : forall l1 l2,
all_equiv_under l1 l2 <-> forall x, In x l1 -> x -> all_equiv l2.