Let $\mathrm{NBG}^-$ be $\mathrm{NBG}$ minus the Axiom of Choice for Classes (including sets)). Further let $\mathrm{DC}$ be the Axiom of Dependent Choice for sets and $\mathrm{DC}^\omega$ be Bernays class form of the Axiom of Dependent Choice.
It has long been known that $\mathrm{NBG}^{-} + \mathrm{DC}$ is a conservative extension of $\mathrm{ZF} + \mathrm{DC}$. Is this a folk theorem or is there a reference for it? Similar question for $\mathrm{NBG}^- + \mathrm{DC}^\omega$ in place of $\mathrm{NBG}^- + \mathrm{DC}$.
Edit (5/3/22)
In his Choice Functions of Sets and Classes (in Sets and Classes, edited by G. H. Müller, North -Holland, 1976, pp. 217-255), Ulrich Felgner shows that in ${\rm NBG}^-$, DC is equivalent to $\mathrm{DC}^\omega$ (though they are not equivalent in the absence of the Axiom of Foundation). As such, the question of folk theorem vs reference reduces to a question about $\mathrm{NBG}^- + \mathrm{DC}$.
For a statement of $\mathrm{DC}^\omega$, see the comments.
