<p>An explicit critical criterion for Flip-Hopf-Hopf bifurcation of discrete systems is proposed for detecting the existence of the codimension-three bifurcation. The explicit critical criteria including eigenvalue assignment, transversality condition and non-resonance condition are established based on the properties of the coefficients of characteristic polynomial equation, which is not involved with the computation of eigenvalues of the linearization matrix. The equivalence between the proposed criterion and the corresponding classical criterion is proved. The map is reduced to a five-dimensional map by the central manifold method. The expressions for the coefficients of the normal form are derived in detail to obtain the normal for corresponding to the reduced five-dimensional map. A three-degree-of-freedom vibro-impact system is taken as an example to show the effectiveness of the proposed explicit criterion and the local dynamical behaviors near bifurcation point.</p>