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We performed temperature- and doping-dependent high-resolution Raman spectroscopy experiments on YBa₂Cu₃O_7−δ to study B_1g phonons. The temperature dependence of the real part of the phonon self-energy shows a distinct kink at T=T_B1g above Tc due to softening, in addition to the one due to the onset of the superconductivity. T_B1g is clearly different from the pseudogap temperature with a maximum in the underdoped region and resembles charge density wave onset temperature, TCDW. We attribute the B_1g phonon softening to an energy gap on the Fermi surface induced by a charge density wave order, which is consistent with the results of a recent electronic Raman scattering study.