Strong subadditivity from relative-entropy monotonicity ($D({\rho }_{ABC}\Vert {\rho }_A\otimes {\rho }_{BC})\ge D({\rho }_{AB}\Vert {\rho }_A\otimes {\rho }_B)$)

Apply data-processing inequality for quantum relative entropy to ${\rho }_{ABC}$ and ${\rho }_A\otimes {\rho }_{BC}$, tracing out $C$. The resulting comparison is $D({\rho }_{ABC}\Vert {\rho }_A\otimes {\rho }_{BC})\ge D({\rho }_{AB}\Vert {\rho }_A\otimes {\rho }_B)$. Expand the logarithms of the product marginals to obtain $S(AB)+S(BC)-S(B)-S(ABC)\ge 0$, exactly Strong subadditivity of Von Neumann entropy.