I've noticed that most of the work in Ackermann set theory is primarily concerned with constructing sets in $V$, the rest of the classes are just excess material, carrying no comprehension over them. There is a try of Muller in which he strengthen the class existence principle of Ackermann into Separation over classes, the resultant theory is $A$, and adding Regularity $R$, and Choice $C$, he gets into $ARC$, a theory claimed [see here] to serve as a foundation of both category and set theory, and thus for most of mathematics.
This gave me the idea of reflecting-out of $V$ principle, since Ackermann's set theory can be interpeted in systems using reflection [see here] , so if to any of the two systems appearing in that posting (with reflection in them re-named as reflection in $V$), we add the following principle:
Reflection out of $V$ schema: if $\varphi$ is a sentence in $FOL(=,\in)$, i.e. doesn't use the symbol $V$, and $\varphi^V$ is the bounded by $V$ sentence of $\varphi$, i.e. the sentence obtained by merely bounding every quantifier in $\varphi$ by $V$, then: $ \varphi^V \to \varphi $, is an axiom.
In other words we are reversing the reflection process, so we are concluding things about classes in general by reflecting from the inside of $V$ to outside it. By that, all set axioms (i.e. sentences in the language of set theory that are satisfied in $V$), would generalize over all classes. This way we easily get to interpret Muller's theory.
Question: is there an obvious inconsistency with a theory that both uses reflection in $V$ and reflection out of $V$ principles?
