Let $M$ be a transitive ground model and let $B \in M$ be a complete Boolean algebra. Let $\varphi(\tau_1,\dots,\tau_n)$ be a formula with parameters from $M^B$. For $b \in B^+ = B \setminus \{0\}$, define $b \Vdash_B \varphi(\tau_1,\dots,\tau_n)$ iff $b \le \|\varphi(\tau_1,\dots,\tau_n)\|_B$. Then $\Vdash_B$ satisfies the [forcing theorem](/theorems/6531): for every $M$-generic ultrafilter $U \subset B$, truth in the quotient $M^B/U$ is exactly captured by membership of Boolean truth values in $U$, and every formula has a Boolean truth value in $B$.