Robinson's joint consistency theorem

Let $T_1$ and $T_2$ be first-order theories. If $T_1$ and $T_2$ are consistent and the intersection $T_1\cap T_2$ is complete (in the common language of $T_1$ and $T_2$), then the union $T_1\cup T_2$ is consistent. Note that a theory is complete if it decides every formula, i.e. either $T \vdash \varphi$ or $T \vdash \neg\varphi$.
Let $T_1$ and $T_2$ be first-order theories. If $T_1$ and $T_2$ are consistent and if there is no formula $\varphi$ in the common language of $T_1$ and $T_2$ such that $T_1 \vdash \varphi$ and $T_2 \vdash \neg\varphi$, then the union $T_1\cup T_2$ is consistent.