Bivariate normal CFD approximation using characteristic function

Question

The normal distribution CFD can be approximated using

$$F_X (x)=P[X≤x]=\frac{1}{2}-\frac{1}{π} \int^{\infty}_{0}\operatorname{Re}\left[\frac{e^{-iux}\phi_X (u)}{iu}\right]du$$

where the characteristic function is given by

$$ϕ_X (t)=e^{iμt-\frac{σ^2 t^2}{2}}$$

I confirmed the results using:

https://www.wolframalpha.com/input?i=standard+normal+cdf+calculator&assumption=%7B%22F%22%2C+%22NormalProbabilities%22%2C+%22z%22%7D+-%3E%221%22

and

https://www.wolframalpha.com/input?i=1%2F2-1%2Fpi*Integrate%5BReal%5B%28e%5E%28-i*u*1%29++e%5E%28-%281%29%5E2++u%5E2%2F2%29%29%2F%28iu%29%5D%2Cu%2C0%2Cinfintity%5D

The charactereistic function for a bivariate normal distribution is given by

$$ϕ(t_1,t_2 )=e^{i(t_1 μ_1+t_2 μ_2)-\frac{1}{2} (σ_1^2 t_1^2+2ρσ_1 σ_2 t_1 t_2+σ_2^2 t_2^2)}$$

https://mathworld.wolfram.com/BivariateNormalDistribution.html

My question is whether a similar approximation for the CFD of a bivariate normal distribution exists, one which specifically makes use of the characteristic function? Ideally, an identity that will show how the characteristic function can be used to determine the CDF.

I have been looking, but couldn't find anything. Thanks