 Solution

Fix forcing names $\tau ,{\tau }_1,\dots ,{\tau }_n\in M$ for the set $w$ and the parameters. Define a nameOnly $\eta$ appearing in $\tau$ and $p\in \mathbb{P}$ are needed, so this is a subset of a ground-model set. The forcing definability lemma makes its defining predicate a formula of $M$. Ground-model axiom schema of separation therefore gives $\sigma \in M$.

If $(\eta ,p)\in \sigma$ with $p\in G$, the accompanying $q\ge p$ belongs to $G$, so ${\eta }_G\in {\tau }_G$. The forcing theorem gives $\phi ({\eta }_G,{\tau }_G,({\tau }_1{)}_G,\dots ,({\tau }_n{)}_G)$.

Conversely, if $u\in {\tau }_G$ satisfies this formula, choose $(\eta ,q)\in \tau$ with $q\in G$ and ${\eta }_G=u$. The forcing truth lemma supplies $r\in G$ forcing the formula for these names. Directedness gives $p\in G$ with $p\le q,r$. Then $(\eta ,p)\in \sigma$, and $u\in {\sigma }_G$. ThusThis proves the instance of the axiom schema of separation in the generic extension, without assuming that instance there in order to construct the name.