Confidence interval difference for the difference of two estimators

Let us assume that we havo two parameters $\alpha$ and $\beta$ and the two maximum likelihood estimators $\hat{\alpha}$ and $\hat{\beta}$. We have also the confidence interval of these estimators, standard error and the covariance is known. How can I derive the confidence interval of the difference $\hat{\alpha}$ - $\hat{\beta}$? I think there is a method called Delta method for this, but I'm not sure and I don't know well how it works. Thank you for anyone able to help me

Given MLEs $\hat{\alpha}$ and $\hat{\beta}$ and their corresponding standard errors, $\mbox{SE}[\hat{\alpha}]$ and $\mbox{SE}[\hat{\beta}]$, approximate 95% Wald-type CIs for $\alpha$ and $\beta$ can be obtained with $$\hat{\alpha} \pm 1.96 \mbox{SE}[\hat{\alpha}]$$ and $$\hat{\beta} \pm 1.96 \mbox{SE}[\hat{\beta}].$$ If you also know the covariance between $\hat{\alpha}$ and $\hat{\beta}$ (let's denote this $\mbox{Cov}[\hat{\alpha}, \hat{\beta}]$), then an approximate 95% Wald-type CI for $\alpha - \beta$ can be obtained with $$(\hat{\alpha} - \hat{\beta}) \pm 1.96 \sqrt{\mbox{SE}[\hat{\alpha}]^2 + \mbox{SE}[\hat{\beta}]^2 - 2\mbox{Cov}[\hat{\alpha}, \hat{\beta}]}.$$