Intersubband lasing at 12-16 microm based on a CO2 laser pumped stimulated resonant Raman process in GaAs/AlGaAs three-level double-quantum-well structures is reported. The presence, or lack of, lasing action provides evidence for resonantly coupled modes of collective electronic intersubband transitions and longitudinal optical phonons. An anticrossing behavior of these modes is clearly seen when the difference between the pump and lasing energies (i.e., Stokes Raman shift) is compared with the subband separation. This work reveals the significance of the strong coupling between intersubband transitions and phonons and raises a new possibility of realizing a phonon "laser."