Let $A$ be a $C^*$-algebra in which $B$ is a $C^*$-subalgebra and $I$ is a closed ideal. In several books on $C^*$-algebras I have encountered the following:
$(B+I)/I$ is $*$-isomorphic to $B/(B\cap I)$.
It seems important, but none of the books I read gives a hint why this is important.
So what does this isomorphism say actually?