= Existential clause of syntactic forcing
For the usual recursive <syntactic forcing relation>, $p\Vdash^*\exists x\,\varphi(x)$ means that conditions $q\leq p$ forcing $\varphi(\sigma)$ for some <forcing name> $\sigma$ are <dense below a forcing condition> $p$. This density clause proves the existential step of the <forcing theorem> from the truth lemma for each named instance. A <generic filter> containing $p$ meets that dense set after it is augmented by conditions incompatible with $p$.
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