= Solution
{c}
We prove the implication from existence of a witness in the extension to a condition forcing the existential statement. If $M[G]\models\exists x\,\varphi(x)$, choose a witness $a\in M[G]$ and a <forcing name> $\sigma\in M$ with $\operatorname{val}(\sigma,G)=a$. The assumed instance of the truth lemma supplies $p\in G$ with
$$
M\models p\Vdash^*\varphi(\sigma).
$$
The <syntactic forcing relation> is monotone under stronger conditions, so every $q\leq p$ 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 gives
$$
\boxed{M[G]\models\exists x\,\varphi(x)\quad\Longrightarrow\quad\exists p\in G\;M\models p\Vdash^*\exists x\,\varphi(x).}
$$
Only the truth lemma assumed for individual named instances and the recursive existential clause were used.
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