Consequences of proposition with existential quantifier

Question

Independent of axioms you are choosing for constructing reals, the following axiom (or theorem) will be true: $\forall a\not=0, \exists a^{-1}, a\cdot a^{-1} = 1$. So, basicaly, this axiom allows us to divide (multiply by inverse) by all numbers except $0$. And I intuitively understand how it does this, but the question which arises for me is: "Are we allowed to use in well-formed formulas only objects which exist ?". Because, it is the only way, I see, we can restrict us (using this axiom) from dividing by zero.