Power Spectral Density of a WSS Random Process with 2D Discrete Fourier Transform

I'm trying to understand how the random process (RP) signal in discrete time domain is related to its power spectral density (PSD) in frequency domain if the signal is wide sense stationary (WSS). If an RP is WSS, then it has the following properly,

$$R_X(n_1,n_2)=R_X(n_2-n_1)=R_X(\tau)$$

So, if RP is WSS and 2D discrete Fourier Transform is completed on the autocorrelation of the RP to get the PSD, why is the below expression true?

$$S_X(\omega_1,\omega_2)=\sum_{n_1}\sum_{n_2}R_X(n_1,n_2)e^{-j\omega_1n_1}e^{-j\omega_2n_2}=S_X(\omega_1+\omega_2)$$

My initial thought was that since the autocorrelation changes from 2D to 1D if WSS, the same should apply in the frequency domain for the PSD. But I do not have the knowledge to go any further than that for proving the above equation to be true.