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If $G$ is a generic filter over a countable transitive model $M$, the generic extension $M[G]$ consists of the interpretations by $G$ of all forcing names in $M$.

Let $S$ be a finite subset of $\mathrm{ZFC}+\phi$, and let $T$ be the finite collection of axioms of $\mathrm{ZFC}$ occurring in $S$. Choose the finite fragment $T^{\ast }$ supplied by the hypothesis. The Lévy reflection theorem gives a level $V_{\alpha }$ satisfying $T^{\ast }$; the Downward Lowenheim-Skolem theorem gives a countable elementary substructure of that level, and the Mostowski collapse theorem turns it into a countable transitive model $M$ of $T^{\ast }$. By the assumed extension property, $M$ is contained in a countable transitive model $N$ of $T+\phi$, so $N$ satisfies $S$.

This argument is formalizable over $\mathrm{ZFC}$ for each finite $T$. Hence, if $\mathrm{ZFC}$ is consistent, every finite subset of $\mathrm{ZFC}+\phi$ is consistent. The compactness theorem now gives

Solved by gpt-5.6-sol high.