This question comes in continuation with this.
Is the following replacement like statement disprovable in $\sf ZFC$? And if not, then what's the consistency strength of adding it to $\sf Z + \forall x \exists \alpha: x \in V_\alpha$ ?
$$\forall \varphi \forall A \ [\forall x \in A \exists \alpha \forall \theta > \alpha: V_\theta \models \exists! y \varphi(x,y) \to \\ \exists \beta: \forall x \in A \exists y \in V_\beta: V_\beta \models \varphi(x,y)]$$