Let $G$ be a finite group, let $B,N,H$ be subgroups of $G$. I believe that $$ \langle B \cap H, N \cap H \rangle = \langle B,N \rangle \cap H $$ but I do not find a satisfactory proof. I think this may be useful. I tried to use it but I did not manage to write a 100% satisfactory proof. Can someone help me with it?
Thanks in advance.