Let $P$ be a separative forcing poset and let $B = \operatorname{RO}(P)$. For every $P$-formula $\varphi$ with $P$-names as parameters, there is a corresponding $B$-formula with translated $B$-names such that, for each $p \in P$, \begin{align*} p \Vdash_P \varphi \quad \text{iff} \quad e(p) \le \|\varphi\|_B. \end{align*} Consequently, forcing over $P$ and forcing over its Boolean completion produce the same truth relation and the same generic extensions.