Suppose that $T$ is a consistent first order theory. Now let the language of $T$ be $L_T$.
Question: is it always consistent to add a new primitive constant $D$, and a new primitive binary relation $\in^*$ called 'class membership', and add a new symbol $\epsilon$ and axiomatize that for each formula $\phi$ the string $\epsilon \phi$ is a term of the language, and add the following schema to all axioms of $T$ bounded $\in^* D$
If $\phi$ is a formula in $L_T$ in which all and only $y,w_1,..,w_n$ occur free, and only occur free, and if $\phi^D$ is the bounded $ \in^* D$ form of the formula $\phi$, then:
$ \forall w1,..,wn \in^* D \ \forall y (y \in^* \epsilon (\forall y\phi^D )\leftrightarrow \phi^D)$,
is an axiom.