For the nonlinearity distortion problem of Mach-Zehnder modulator (MZM) applied in the on-board microwave photonics system, the situation for two input radio frequency (RF) signals with different frequencies and phases is discussed, and an exact analytical solution is derived with the method of expanding Bessel series and Graf addition theory. According to the analytical expression, the nonlinearity characteristics of the modulator can be precisely predicted, and the system performance can be optimized. The correctness of the analytical solution is approved by simulation results. Analytical results indicate that the nonlinearity distortion is suppressed as the decrease of modulation index, the increase of direct current bias phase shift and phase difference between two input RF signals. When the phase difference equals zero or π and the direct current bias phase shift is π/2, there are only odd-order distortion terms. When the phase difference equals zero or π and the direct current bias phase shift is π, there are only even-order distortion terms.