  Ever since the introduction of behavioral equivalences on processes one has
been searching for efficient proof techniques that accompany those
equivalences. Both strong bisimilarity and weak bisimilarity are accompanied by
an arsenal of up-to techniques: enhancements of their proof methods. For
branching bisimilarity, these results have not been established yet. We show
that a powerful proof technique is sound for branching bisimilarity by
combining the three techniques of up to union, up to expansion and up to
context for Bloom's BB cool format. We then make an initial proposal for
casting the correctness proof of the up to context technique in an abstract
coalgebraic setting, covering branching but also {\eta}, delay and weak
bisimilarity.
