$\mathcal{Z}\models \phi(13)$ if and only if $\mathcal{Z}\models\phi(-13)$

Question

Consider the structure $\mathcal{Z}=(\mathbb{Z},0,+,-)$ in the language of abelian groups $\mathcal{L}_{ab}=(0,+,-)$. we want to show that $\mathcal{Z}\models \phi(13)$ if and only if $\mathcal{Z}\models\phi(-13)$

I don't know how to proceed. do we do it by induction on the complexity of the formula $\phi$?

Answer 1

The function $f(n) = -n$ is an automorphism of $\mathcal{Z}$. Since isomorphisms preserve satisfaction of formulas, for any formula $\varphi(x)$, $\mathcal{Z}\models \varphi(13)$ if and only if $\mathcal{Z}\models \varphi(f(13))$ if and only if $\mathcal{Z}\models \varphi(-13)$.