Group of line bundles with $\otimes$ isomorphic to $H^1(B,\mathbb{Z}_2)$

Question

I want to prove that all line bundles with the same base $B$ form group with operation $\otimes$ which isomorphic to $H^1(B,\mathbb{Z}_2)$. It's obvious to me, that there is bijection between line bundles and $H^1(B,\mathbb{Z}_2)$ which exists because $B\mathbb{Z}_2 = K(\mathbb{Z}_2,1)$ but I dont understand why that group is isomorphism. I found some proof in icc prop. 3.7.12 but it uses concept of Euler characteristic class, and I want to some prove which uses only concept of Stiefel Whitney class, does that exist? Hope for your help.