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In non-hadronic axion models, which have a tree-level axion-electron interaction, the Sun produces a strong axion flux by bremsstrahlung, Compton scattering, and axiorecombination, the "BCA processes." Based on a new calculation of this flux, including for the first time axio-recombination, we derive limits on the axion-electron Yukawa coupling gae and axion-photon interaction strength ga using the CAST phase-I data (vacuum phase). For ma <~ 10 meV/c2 we find ga gae < 8.1 × 10−23 GeV−1 at 95% CL. We stress that a next-generation axion helioscope such as the proposed IAXO could push this sensitivity into a range beyond stellar energy-loss limits and test the hypothesis that white-dwarf cooling is dominated by axion emission.