With the LHC experiments gathering more data, the exploration of the symmetry breaking sector of the Standard Model will gain renewed impetus. Likewise, it is important to search for dark matter candidates being a degree of freedom missing in the Standard Model. An invisible axion is an interesting candidate for dark matter. However trying to look for direct evidence of its existence at the LHC is hopeless as it is extremely weakly coupled. Therefore we have to resort to less direct ways to explore this sector by formulating consistent models that include the axion and deriving consequences that could be experimentally tested. In this talk we explored such consequences in the DFS model, an extension of the 2HDM with axion.