To quote:

It is well known that, if $W_1, ..., W_n, Z_1, ..., Z_m$ are random variables and $a_1, ..., a_n, b_1, ..., b_m$ are constants, then

$Cov ( \sum_{i=1}^n a_iW_i, \sum_{j=1}^m b_jZ_j) = \sum_{i=1}^n \sum_{j=1}^m a_i b_j Cov(W_i,Z_j)$.

Use this property (and other properties of covariance) to prove that for the model $Y_i = \alpha + \beta x + \epsilon$, with $\epsilon \sim N(0,\sigma^2)$, we have $Cov(\hat{\beta},\bar{Y}) = 0 $,

where $\hat{\beta} = S_{XY}/S_{XX}$.