Let be any Language.
Let and be -Structures
such that is a Substructure of .
Let be a set of -formulas closed under subformulas.
Then all formulas in are Absolute between and
if and only if
for any formula where has Free Variables including
and for any such that
there is some such that
Proof
By induction on formula complexity.
We only need to check formulas where is Absolute.
Note that this is already Upwards Absolute.
Now suppose that
for some .
By assumption, there is some such that