I have proven that if the systems $\mathcal G$ and $\mathcal H$ are independent then so are the Dynkin systems $\delta(\mathcal G)$ and $\delta(\mathcal H)$. Now I'd like to generalize it to $n$ systems yet I don't quite see how to finish the proof by induction.
Say, $\mathcal G_1, \ldots, \mathcal G_n $ are independent and by the induction hypothesis we know that each $(n-1)$ of $\delta(\mathcal G_1), \ldots, \delta(\mathcal G_n) $ are independent. Thus we only need to show that $$E_1 \in\delta(\mathcal G_1), \ldots, E_n \in \delta(\mathcal G_n) \implies \mathbb P \left(\bigcap_{j=1}^n E_j\right)= \prod_{j=1}^n \mathbb P (E_j).$$
You may prove by induction that $\delta(\mathcal{G}_1),\dots,\delta(\mathcal{G}_k),\mathcal{G}_{k+1},\dots,\mathcal{G}_n$ are independent (for all $k\le n$)...
For $k=0$ this is the assumption ($\mathcal{G}_{1},\dots,\mathcal{G}_n$ are independent). For $k>1$ assume that $\delta(\mathcal{G}_1),\dots,\delta(\mathcal{G}_{k-1}),\mathcal{G}_k,\dots,\mathcal{G}_n$ are independent. Let $F=\bigcap_{i\ne k}G_i$ for sets $G_i\in \delta(\mathcal{G}_i)$ if $i<k$ and $G_i\in\mathcal{G}_i$ if $i>k$, and let
$$\mathcal{E}_F=\{G\in \delta(\mathcal{G}_k):P\{G\cap F\}=P\{G\}P\{F\}\}$$
$\mathcal{E}_F$ is a Dynkin system s.t. $\delta(\mathcal{G}_k)\subset \mathcal{E}_F$ for every $F$ as defined above (because $\mathcal{G}_k\subset \mathcal{E}_F$ for every $F$) which implies the result (notice that $P\{F\}=\prod_{i\ne k}P\{G_i\}$).