We study the manifestation of the ∆ ++ − ∆ − component of the deuteron wave function in the exclusive reaction pd → π − π − ∆ ++ . Due to the large binding energy the internal motion in the ∆ − ∆ system is relativistic. We take this into account within the light-cone (LC) wave function formalism and, indeed, found large differences between calculations based on the LC and nonrelativistic (NR) wave functions. We demonstrate, that the consistent LC treatment of the ∆ − ∆ system plays the key role in the separation of the signal and background. Within the LC approach, the characteristic shape of the momentum distribution of the ∆−∆ bound system predicted by the meson-exchange model is well visible on the background of usual annihilations at beam momenta between 10 and 15 GeV/c.