I need a clarification about Poisson brackets.
I'm studying on Goldstein's Classical Mechanics (1 ed.).
Goldstein proves that Poisson brackets are canonical invariants for any functions F and G.
But there is a step that I can't understand.
After some steps, he says that:
$$ \tag{1} [F, G]_q,_p = \sum_k ( \frac { \partial G}{\partial Q_k} [F,Q_k]_q, _p +\frac {\partial G}{\partial P_k}[F, P_k]_q, _p)$$
After other steps, he writes:
$$ \tag{2}[F,Q_k]= - \frac {\partial F}{\partial P_k}$$
and $$ \tag{3}[P_k, F]_q, _p = \sum_j \frac {\partial F}{\partial Q_j} [P_k, Q_j] + \sum_j \frac {\partial F}{\partial P_j}[P_k, P_j]$$
$$\Rightarrow \tag {4} [F,P_k]=\frac {\partial F}{\partial Q_k}$$
and now he replaces these relations in the first expression I have written, obtaining:
$$\tag {5}[F, G]_q, _p=[F, G]_Q, _P$$
Why does he obtain in the second last step $\frac {\partial F}{\partial Q_k}$ and not $-\frac {\partial F}{\partial Q_k}$? $[P_k, F]=-[F, P_k]$ isn't it?
EDIT: Golstein starts from (1) and substituites $Q_k$ to $F$ and $F$ to $G$ and so he obtains (2).
Then he substitutes $P_k$ to $F$ and $F$ to $G$ and obtains (3).
Immediately after he writes (4), that according to me is opposite to (3).
And so I have thought to a printing error. I have tried to substitute $-P_k$ to $F$ and I have obtained
$$[-P_k, F]=\frac {\partial F}{\partial Q_k}$$
Then, as $[-P_k, F]=[F,P_k]$, I can say that $$ [F,P_k]=\frac {\partial F}{\partial Q_k}$$
And so I can obtain (5).
Could you confirm that my argumentation is correct?