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1. Extensionality: as in ZF
2. Class comprehension schema: as in Morse-Kelley
3. The empty class is a set
4. Singletons: $\forall x [ set(x) \to set(\{x\})]$
5. Boolean Union: $\forall x,y [set(x) \wedge set(y) \to set (x \cup y)]$

Now it is clear that all axioms of Union, Power and Separation over sets are derivable from the above transfer principle, and so $ZC$ is interpretable here. Actaully if we restrict $\phi$ to have no more that three atomic subformulas, we can still interpret the whole of $ZC$. Replacement is not interpretable by this principle. Yet, a minor modification of this principle might succeed in proving replacement over sets, this can be done by changing the closure property to involve only subsets of $HF$, what I call as "proximity closure over HF", so to restate that:

1. Extensionality: as in ZF
2. Class comprehension schema: $\forall x_1,..,x_n\ \exists x \ [x=\{y|set(y) \wedge \phi(y,x_1,..,x_n)\}] $
3. The empty class is a set
4. Singletons: $\forall x [ set(x) \to set(\{x\})]$
5. Boolean Union: $\forall x,y [set(x) \wedge set(y) \to set (x \cup y)]$

Now it is clear that all axioms of Union, Power and Separation over sets are derivable from the above transfer principle, and so $ZC$ is interpretable here. Actaully if we restrict $\phi$ to have no more than three atomic subformulas, we can still interpret the whole of $ZC$. Replacement is not interpretable by this principle. Yet, a minor modification of this principle might succeed in proving replacement over sets, this can be done by changing the closure property to involve only subsets of $HF$, what I call as "proximity closure over HF", so to restate that:

I don't have any proof of consistency of these principles, but if there is no clear inconsistency of those relative to $ZF$ or $MK$ or some extension of those, then could it be possible to think of extending that principle to properties other than "x is finite"? so we generalize it to any property $P$, that is:

Of course in both cases $Q$ must be expressible by a formula strictly shorter than the shortest expression defining property $P$, and also we stipulate parallel axioms sufficient to define the property $P$, also axioms asserting the element-hood of all hereditarily $P$ classes, and the existence of a set of all hereditarily $P$ sets. Of course this can only be done for some selected properties $P$.

is that possible or it is involved with clear inconsistencies? and what would be the general qualification of such property $P$

I don't have any proof of consistency of these principles, but if there is no clear inconsistency of those relative to $ZF$ or $MK$ or some extension of those, then could it be possible to think of extending that principle to properties other than "x is finite"? so we generalize it to some line properties, so for a property $P$ in that line, we stipulate that:

Of course in both cases $Q$ must be expressible by a formula strictly shorter than the shortest expression defining property $P$, and also we stipulate parallel axioms sufficient to define the property $P$, also axioms asserting the element-hood of all hereditarily $P$ classes, and the existence of a set of all hereditarily $P$ sets. Of course this can only be done for some selected line of properties.

is that possible or it is involved with clear inconsistencies? and what would be the general qualification of such property $P$?

$\forall x [HF(x) \to \forall y (\phi(y,x) \to HF(y))] \to \forall x \in V [ \forall y (\phi(y,x) \to y \in V)]$

$\forall x [HF(x) \to \forall y \subseteq HF (\phi(y,x) \to HF(y))] \to \forall x \in V [ \forall y (\phi(y,x) \to y \in V)]$

$\forall x [x \in HF \to \forall y (\phi(y,x) \to y \in HF)] \to \forall x \in V [ \forall y (\phi(y,x) \to y \in V)]$

$\forall x [x \in HF \to \forall y \subseteq HF (\phi(y,x) \to y \in HF)] \to \forall x \in V [ \forall y (\phi(y,x) \to y \in V)]$