They didn't. Much of the rhetoric they used, if anything, was about the belief they were fighting for its ideals which the North was trampeling on. /u/freedmenspatrol writes extensively about this in this older post.

In the song “I’m a good ol rebel” recorded by Hoyt axton, there’s a line about how he hates the Declaration of Independence. And I was watching a YouTube video that suggested that the DoI wasn’t always the sacred document that it is now. It got popular in the civil war era after republicans used the “all men created equal” to justify their cause.