How is equivalence of axioms and statements treated in the foundations of mathematics?

In classical logic any true statements meaning any theorems and axioms are equivalent to each other, however this does not offer us much information about their actual connection. Taking ac for example its famously equivalent to Zorns lemma for example, but in zfc it would also be equivalent to "1+1=2" for example. By equivalence is what is meant then in just zf ac which cannot be proven in zf can be shown to be equivalent to Zorns lemma? For some reason I just thought about this in a way like basis and linear dependence in vector spaces. Do you think this analogy makes sense or were does it stop working?