 Solution

We prove the implication from existence of a witness in the extension to a condition forcing the existential statement. If $M[G]\models \exists x\phi (x)$, choose a witness $a\in M[G]$ and a forcing name $\sigma \in M$ with $\mathrm{val}(\sigma ,G)=a$. The assumed instance of the truth lemma supplies $p\in G$ withThe syntactic forcing relation is monotone under stronger conditions, so every $q\le p$ also forces this same named instance. In particular, the conditions below $p$ having some forced witness are dense below a forcing condition $p$. The existential clause of syntactic forcing therefore givesOnly the truth lemma assumed for individual named instances and the recursive existential clause were used.