In the Minimal Standard Model (MSM) there is no degree of freedom for dark matter. There are several extensions of the MSM introducing a new particle - an invisible axion, which can be regarded as a trustworthy candidate at least for a part of the dark matter component. However, as it is extremely weakly coupled, it cannot be directly measured at the LHC. We propose to explore the electroweak sector indirectly by considering a particular model that includes the axion and derive consequences that could be experimentally tested. We discuss the Dine-Fischler-Srednicki (DFS) model, which extends the two-Higgs doublet model with an additional Peccei-Quinn symmetry and leads to a physically acceptable axion. The non-linear parametrization of the DFS model is exploited in the generic case where all scalars except the lightest Higgs and the axion have masses at or beyond the TeV scale. We compute the oblique corrections and use their values from the electroweak experimental fits to put constraints on the mass spectrum of the DFS model.