Conditional Covariance is zero in classical linear regression model

Question

I just read that in the classical linear regression model (Y=Xβ+ε) the Cov(β ̂,ε ̂│X)=0.

How can we derive this fact?

What is clear is that if X and Y are independent, then Cov(X,Y)=0.

Also, for any constants a1, b1, a2, and b2,

Cov(a1X + b1, a2Y + b2) + a1 a2 Cov(X,Y).

And E(Y|X) + E(Y). Also,

Var(Y|X) = Var(Y)