As an example, when set theory was created, it seems people tried to write the mathematical concepts they had using set theory, I guess that after this, what were previously axioms became theorems in set theory.
I've made a silly example: Suppose we have $aRa$ as an axiom. I guess that we could find a formal system which have these relations: $\{\circ,\bullet\}$ and products between them: $\{\circ\circ,\circ\bullet,\bullet\bullet\}$ and $$\circ\bullet:=R$$
So now when we have:
$$a\circ \bullet a$$
$$aRa \tag{$\circ\bullet:=R$}$$
My question may be rather vague because I employed only my gut feeling about formal systems (and also some naive readings about it).
An axiom of a formal system is a theorem of that system. That is, any axiom proves itself.
But I suppose you want a different formal system, which doesn't have the given statement as an axiom. Well, OK, replace axiom $A$ by $A \; \text{or}\; F$, where $F$ is any statement that can can be proven to be false (without use of $A$).