  We demonstrate that chiral symmetry breaking occurs in the confining phase of
QCD-like theories with $N_c$ colors and $N_f$ flavors. Our proof is based on a
novel strategy, called `downlifting', by which solutions of the 't Hooft
anomaly matching and persistent mass conditions for a theory with $N_f-1$
flavors are constructed from those of a theory with $N_f$ flavors, while $N_c$
is fixed. By induction, chiral symmetry breaking is proven for any $N_f\geq
p_{min}$, where $p_{min}$ is the smallest prime factor of $N_c$. The proof can
be extended to $N_f <p_{min}$ under the additional assumption on the absence of
phase transitions when quark masses are sent to infinity. Our results do not
rely on ad-hoc assumptions on the spectrum of massless bound states.
