Given a set Ω with a σ-field F defined on it. Let B be a...

Suggested proof. (Axioms are $\sigma$-algebra axioms.) Notation different from question.
Let ($\star$) be
$B\cap \mathcal{E}:=\{B\cap E,E\in \mathcal{E}\}={\mathcal{E}}_B$
To prove the claim it is a $\sigma$-algebra on $B$ we need to show:
(i) $B\in {\mathcal{E}}_B$
(ii) $(B\cap E{)}^c$ in ${\mathcal{E}}_B$ $\mathrm{\forall }E\in \mathcal{E}$.
(iii) ${\mathcal{E}}_B$ satisfies $\sigma$-additivity.
Proof of (i): We know that $\mathrm{\Omega }$ is in $\mathcal{E}$ by Axiom (1).
Therefore, by ($\star$), $(B\cap \mathrm{\Omega })\in {\mathcal{E}}_B$.
But $B\cap \mathrm{\Omega }=B$, hence $B\in {\mathcal{E}}_B$.
Proof of (ii): $(B\cap E{)}^c=(B^c\cup E^c)$ (de Morgan).
Now $B^c$ and $E^c$ are in $\mathcal{E}$ by Axiom (2) and their union is in $\mathcal{E}$ by Axiom (3).
Hence, by ($\star$) $B\cap (B^c\cup E^c)\in {\mathcal{E}}_B$.
But $(B^c\cup E^c)=(\mathrm{\Omega }\cup \mathrm{\Omega })=\mathrm{\Omega }$, and $B\cap \mathrm{\Omega }=B$ which is in ${\mathcal{E}}_B$ by (i).
Proof of (iii): Let $E_j\in \mathcal{E}$.
We want to show that
$\bigcup _{j\in \mathcal{N}}(B\cup E_j)\in {\mathcal{E}}_B.$

But,
$\bigcup _{j\in \mathcal{N}}(B\cup E_j)=B\cap \bigcup _{j\in \mathcal{N}}{E}_J$$\bigcup _{j\in \mathcal{N}}{E}_J=X,\text{ say, then }X\in \mathcal{E}$
by Axiom (3).
Hence, by $(\star )$ $(B\cap X)\in {\mathcal{E}}_B$