In my homological algebra notes I found the proof that the homotopy category $K^{-}(\mathcal{P})$ is equivalent to the derived category $D^{-}(\mathcal{A})$ when $\mathcal{A}$ has enough projectives.
The proof essentially relies on the following:
Lemma: Every quasi isomorphism of $K^{-}(\mathcal{P})$ is an isomorphism.
Lemma: Given a complex $A^{\bullet}$ in the category of complexes $C(\mathcal{A})$ then there exists a complex $P^{\bullet}$ and a quasi isomorphism between $P^{\bullet}$ and $A^{\bullet}$ with $P^{\bullet}$ having projective terms.
I was wondering whether this result could be extended to $K^{b}(\mathcal{P})$ is equivalent to $D^{b}(\mathcal{A})$. I can't quite imply an immediate generalization. Is this a well-known fact?
Any help would be appreciated, any other proof supporting the truthfulness of the fact would be welcomed.