Let $B$ be a complete Boolean algebra in $V$ and $V^B$ be the Boolean-valued universe defined in the standard way. For any $x \in V$, $\check{x}$ is defined as $\{ \langle \check{y}, 1_B \rangle\ : y \in x \}$. When $B$ is atomic, is there a direct way of proving that for every formula $\varphi$, $V \models \varphi(x)$ iff $V^B \models \varphi(\check{x})$ (i.e., the $B$ value of $\varphi(\check{x})$ is $1_B$)?
I know that if we start with a transitive model $M$ of ZFC and consider an atomic complete Boolean algebra $B$ in $M$, then using Boolean-ultrapower we can show that $M$ is an elementary submodel of $M^B$ in the sense above. But I wonder if there is a proof of this fact about $V$ and $V^B$ without going through any transitive model.