Let $T$ be a finite collection of axioms of $\mathrm{ZF}$, let $\sigma$ be a sentence in the language of $\mathrm{ZF}$ and consider the statement

$\tau$: “any transitive countable model of $T$ satisfies $\sigma$”.

Then $\mathrm{ZFC}\vdash\tau$ implies $\mathrm{ZFC}\vdash\sigma$, by the classical argument using Reflection, Downward Löwenheim–Skolem and Mostowki’s collapse to get a countable transitive model where finitely many sentences are absolute.

My question is: does $\mathrm{ZF}\vdash\tau$ imply $\mathrm{ZF}\vdash\sigma$? This may look trivial but, without choice, Downward Löwenheim–Skolem can’t be used as above. On the other hand, it could be argued that $\mathrm{ZF}\vdash$ “there is a proof of $\sigma$ from $T$”, but this does not mean that such a proof can be found in the meta-theory.

Thank you for your help with this (possibly trivial) matter.