A quantitative phase-field (PF) model with an anti-trapping current (ATC) is developed to simulate the dendritic growth with two-sided diffusion. The asymptotic analysis is performed at the second-order for the PF equations coupled with nonlinear thermodynamic properties and an ATC term under the equal chemical potential condition. The PF mobility and ATC are derived based on the asymptotic analysis in the thin interface limit, and the solute drag model. Then the model is reduced to the dilute solution limit for dendrite solidification of binary alloys. The test of convergence with respect to the interface width exhibits an excellent convergent behavior of the proposed model. The performance of the model is then validated by comparing PF simulations with the predictions of the Gibbs-Thomson relation, the linearized solvability theory, and the modified-Lipton-Glicksman-Kurz (M-LGK) analytical model, for the isothermal dendritic growth of an Fe-0.15 mol%C alloy. The results demonstrate quantitative capabilities of the model that effectively suppresses the abnormal solute trapping effect when the interface is taken artificially to be wide. It is also found that the present model can quantitatively describe dendrite growth with various solid diffusivities, ranging from the case with one-sided diffusion to the symmetrical model.