  In this paper, the notion of conditionally bi-free independence for pairs of
algebras is introduced. The notion of conditional $(\ell, r)$-cumulants are
introduced and it is demonstrated that conditionally bi-free independence is
equivalent to mixed cumulants. Furthermore, limit theorems for the additive
conditionally bi-free convolution are studied using both combinatorial and
analytic techniques. In particular, a conditionally bi-free partial
$\mathcal{R}$-transform is constructed and a conditionally bi-free analogue of
the L\'{e}vy-Hin\v{c}in formula for planar Borel probability measures is
derived.
