Wrapping a where clause to preserve rows with nulls

I have a query where I have to run a where clause generated by another
system (i.e., I can't modify that where clause. The where clause may return
`null`, but I actually want to keep rows that return `null` (and rows that
return `true` but not rows that return `false`).

I thought it would be as simple as wrapping in `(...) is not false` but
that seems to prevent index usage.

For example, let's say that given the table:

CREATE TABLE test_index (value INTEGER);
CREATE INDEX idx_value ON test_index(value);

And the predicate `value = 5000`, if I run the original query that excludes
rows where the predicate is null it uses the index:

SELECT *
FROM test_index
WHERE value = 5000;

But as soon as I tack on an `IS NOT FALSE` the index is not used:

SELECT *
FROM test_index
WHERE (value = 5000) IS NOT FALSE;

This was surprising to me. I was hoping this might be able to use the index.

Is there any way to include the rows where the predicate evaluates to null
while still using an index?