quantum field theory - Difference between "C-violation without CP-violation" and "C-violation with CP -violation"

Consider two possible decay channels of a massive particle as $X\to A+B$ and $X\to C+D$ with decay rates $r$ and $1-r$ respectively. Let the decay rates of its antiparticle into channels $\bar X\to \bar A+\bar B$ and $\bar X\to \bar C+\bar D$ are respectively $\bar r$ and $1-\bar r$.

For a theory with C-violation but CP-conservation, although the decay angular distribution for X and $\bar X$ would be different the decay rates integrated over all angles will be equal i.e., $\Gamma_X=\Gamma_{\bar X}$. But for a theory with both C and CP-violation would ensure different absolute rates in the two channels i.e., $\Gamma_X\neq\Gamma_{\bar X}$.

How can I understand/prove this statement? For the reference, see this.