Consider a linear model
$y_i= \alpha + \beta x_i + \epsilon_i$
and estimates for the slope and intercept $\hat{\alpha}$ and $\hat{\beta}$ using ordinary least squares. This reference for a mathematical statistics makes the statement that $\hat{\alpha}$ and $\hat{\beta}$ are independent (in their proof of their theorem).
I'm not sure I understand why. Since
$\hat{\alpha}=\bar{y}-\hat{\beta} \bar{x}$
Doesn't this mean $\hat{\alpha}$ and $\hat{\beta}$ are correlated? I'm probably missing something really obvious here.