What's the consistency strength of adding the following single sentence replacement like statement to $\sf Z + \forall x \exists \alpha: x \in V_\alpha$ ?

$$\forall \varphi \forall A \ [\forall x \in A \exists \alpha \forall \theta > \alpha: V_\theta \models \exists! y \varphi(x,y) \to \\ \exists \beta: \forall x \in A \exists y \in V_\beta: V_\beta \models \varphi(x,y)]$$

The idea is that $\sf ZF$ is not aximatizable by adding a single sentence to axioms of Zermelo set thery, on the other hand the abve statement seems to be a thereom of $\sf ZFC$, so this means that the above theory must be stricty weaker than $\sf ZFC$.

Which of the known fragments of $\sf ZFC$ this theory would be equivalent to?

Note: This question has already been asked on MathStackExchange and it is not answered yet.

What's the consistency strength of adding the following single sentence replacement like statement to $\sf Z + \forall x \exists \alpha: x \in V_\alpha$ ?

$$\forall \varphi \forall A \ [\forall x \in A \exists \alpha \forall \theta > \alpha: V_\theta \models \exists! y \varphi(x,y) \to \\ \exists \beta: \forall x \in A \exists y \in V_\beta: V_\beta \models \varphi(x,y)]$$

The idea is that $\sf ZF$ is not aximatizable by adding a single sentence to axioms of Zermelo set theory, on the other hand the above statement seems to be a theorem of $\sf ZFC$, so this means that the above theory must be strictly weaker than $\sf ZFC$.

Which of the known fragments of $\sf ZFC$ this theory would be equivalent to?

Note: This question has already been asked on MathStackExchange and it is not answered yet.