Since you say the filtered table contains a relatively small number of rows, your best option is to create a separate table with the same columns as the original table A and rows that match your criteria , then replace the original table and delete it. Also, with this number of IDs, you may want to use WHERE IN () instead of a join - as long as the fields you use there are indexed, it will usually be way em> faster. Putting it all together:

CREATE TABLE new_A AS
  SELECT A.* FROM A 
  WHERE classification_id IN (SELECT classification_id FROM B);
RENAME TABLE A TO old_A, new_A to A;
DROP TABLE old_A;

Things to note:

• Back up your data! And test the query thoroughly before running DROP TABLE. You don’t want to lose 57M rows of data because of random answers on StackOverflow.
• If A has any indexes or foreign keys, these will not be copied - so you must recreate them manually. I recommend running SHOW CREATE TABLE A first and taking note of its structure. Alternatively, you might consider using the output of SHOW CREATE TABLE A as a template to explicitly create table new_A, and then do INSERT INTO new_A SELECT ... code> instead of CREATE TABLE new_A AS SELECT ... and use the same query after that.

Try this:

delete from "table A" where classification_id not in (select classification_id from "table B");