# Why the restriction to "elements of $V$' in the output of formulas used in Reflection axiom schema of Ackermann class theory?

Can we upgrade the Reflection axiom schema in Ackermann to the following:

Modified Reflection axiom schema: if $$\psi(y)$$ is a formula that doesn't use the symbol $$V$$, in which only symbols $$y,x_1,..,x_n$$ occur free, and in which $$x$$ is not free, then all closures of:

$$x_1,..,x_n \in V \wedge \forall y (\psi(y) \to y \subset V) \to \\\exists x \in V \forall y (y \in x \leftrightarrow \psi(y))$$; are axioms.

In the traditional exposition of Ackermann set theory, the output of $$\psi(y)$$ is restricted to elements of $$V$$. While here it is eased as to allow subsets of $$V$$.

With the above formulation, there is no need for a second completeness axiom for $$V$$, since it would be redundant. We only need the axiom of heredity, that is the first completeness axiom for $$V$$, which only mounts to saying that the world $$V$$ of all sets is transitive, a very natural statement!

Question: Is there a clear problem with the above scheme?

I tend to think that the above schema is a theorem schema of Ackermann set theory.