That's the wrong way to think about it, I think. It's more correct to think of axioms as being preconditions. We say, "Here is what happens if these things ar

That's the wrong way to think about it, I think. It's more correct to think of axioms as being preconditions. We say, "Here is what happens if these things are true", and then specify what happens, but that doesn't deprive us from independently checking whether those conditions are true. Just because something has been proven does not stop it from being an axiom. It's contextural.