Theorem 9.7 (Lipschiptz principal). Assuming that:

• $\varphi$ an $L_{\text{rings}}$ sentence

Then the following are equivalent:

• (1) ${\mathrm{ACF}}_0\models \varphi$, i.e. $\varphi$ in every $k\models {\mathrm{ACF}}_0$
• (2) ${\mathrm{ACF}}_0\cup \{\varphi \}$ is consistent
• (3) there exists some $n>0$ such that ${\mathrm{ACF}}_p\models \varphi$ for any $p>n$
• (4) for all $n>0$, there exists some $p>n$ such that ${\mathrm{ACF}}_p\cup \{\varphi \}$ is consistent