Some axioms for type theories

The $\lambda\Pi$-calculus modulo theory is a logical framework in which many type systems can be expressed as theories. We present such a theory, the theory $\mathcal{U}$, where proofs of several logical systems can be expressed. Moreover, we identify a sub-theory of $\mathcal{U}$ corresponding to each of these systems, and prove that, when a proof in $\mathcal{U}$ uses only symbols of a sub-theory, then it is a proof in that sub-theory.