The authors derive a single-band effective Hamiltonian for high-$T_c$ Cu-oxide superconductors, assuming that doping primarily creates holes on oxygen sites. They show that the strong hybridization between Cu and O atoms binds a hole on each oxygen site to the central Cu$^{2+}$ ion, forming a local singlet. This singlet moves through the lattice in a manner similar to a hole in the single-band effective Hamiltonian of the strongly interacting Hubbard model. The key point is that two holes feel a strong repulsion against residing on the same square, leading to the recovery of the single-band model. The Hamiltonian is derived starting from a two-band model, and the authors emphasize the importance of phase coherence in producing the large energy separation between different symmetry states and the spin-singlet state of the Cu hole and the symmetric O hole. The effective Hamiltonian is shown to be consistent with the single-band Hubbard model in the large-$U$ limit.The authors derive a single-band effective Hamiltonian for high-$T_c$ Cu-oxide superconductors, assuming that doping primarily creates holes on oxygen sites. They show that the strong hybridization between Cu and O atoms binds a hole on each oxygen site to the central Cu$^{2+}$ ion, forming a local singlet. This singlet moves through the lattice in a manner similar to a hole in the single-band effective Hamiltonian of the strongly interacting Hubbard model. The key point is that two holes feel a strong repulsion against residing on the same square, leading to the recovery of the single-band model. The Hamiltonian is derived starting from a two-band model, and the authors emphasize the importance of phase coherence in producing the large energy separation between different symmetry states and the spin-singlet state of the Cu hole and the symmetric O hole. The effective Hamiltonian is shown to be consistent with the single-band Hubbard model in the large-$U$ limit.