Conditional Functional Dependencies (CFDs) have been recently introduced in the context of data cleaning. They can be seen as an unification of Functional Dependencies (FD) and Association Rules (AR) since they allow to mix attributes and attribute/values in dependencies. In this paper, we introduce our fist results on constant CFD inference. Not surprisingly, data mining techniques developed for functional dependencies and association rules can be reused for constant CFD mining. We focus on two types of techniques inherited from FD inference: the first one extends the notion of agree sets and the second one extends the notion of non-redundant sets, closure and quasi-closure. We have implemented the latter technique on which experiments have been carried out showing both the feasibility and the scalability of our proposition.