Elitzur's theorem can also be generalized to a larger notion of local symmetries where in a D-dimensional space, there can be symmetries that act uniformly on a d-dimensional hyperplanes. In this view, global symmetries act on D-dimensional hyperplanes while local symmetries act on 0-dimensional ones. The generalized Elitzur's theorem then provides bounds on the vacuum expectation values of operators that are non-invariant under such d-dimensional symmetries. This theorem has numerous applications in condensed matter systems where such symmetries appear.