probability of intersection of confidence intervals

Question

Suppose $\hat{\boldsymbol{\theta}} = \left(\hat{\theta}_1,\hat{\theta}_2\right)^{\top} \xrightarrow d N_2(\boldsymbol{\theta}, V)$ for some unknown $V.$ We have consistent estimator of $V:$ $\hat{V},$ and can construct confidence intervals for both $\theta_1$ and $\theta_2$ separately. I would like to calculate the probability that both such pointwise confidence intervals hold (not confidence elipsoid, but rather rectangular setup). Can I do it by using the Cramer-Slutzsky theorem and integrating the bivariate normal density with estimated variance matrix $\hat{V},$ mean zero and bounds quantile times the estimated standard deviation? To get the simultaneous coverage for some chosen confidence level? The confusion stems from the fact that both bounds and density are then random, though we could standardize $\hat{\theta}_1,\hat{\theta}_2$ beforehand. Thank you so much.