Since $E$ is independent in $P$, one has the probability decomposition \begin{equation} \mathbb{P}(E_{j_1} \cap \dots \cap E_{j_n}) = \mathbb{P}(E_{j_1}) \dots \mathbb{P}(E_{j_n}) \end{equation} for any finite number of events $E_{j_1}, \dots, E_{j_n}$ in $E$. Hence, one has the above decomposition for any pairs of events $E_{j_1}, E_{j_2}$ in $E$, and this is precisely the condition for pairwise independence. $\square$