The problem is that double negation elimination is already equivalent to classical logic (i.e. adding double negation elimination to minimal logic gives the law of excluded middle as a theorem), so there is no “background theory” like ZF to work in.
The problem is that double negation elimination is already equivalent to classical logic (i.e. adding double negation elimination to minimal logic gives the law of excluded middle as a theorem), so there is no “background theory” like ZF to work in.