Is the following a theorem of $\sf ZF(C)$?

Countable reflection: If $\phi$ is a formula in which $W$ is not free, then:

$(\phi \to \exists W: |W|= \omega \land \operatorname {Transitive}(W) \land \phi^W)$

Where $\phi^W$ is the formula obtained by merely bounding all quantifiers in $\phi$ by $ \in W$.

That is: every true sentence is reflected upon a countable transitive set.

So all true sentences are reflected upon some elements of $V_{\omega_1}$ (or $V_{\omega_2}$ in case of $\sf ZF$).