  We reconsider the contribution due to $\pi a_1$-mixing to the anomalous
$\gamma\to\pi^+\pi^0\pi^-$ amplitude from the standpoint of the low-energy
theorem $F^{\pi}=e f_\pi^2 F^{3\pi}$, which relates the electromagnetic form
factor $F_{\pi^0\to\gamma\gamma}=F^\pi$ with the form factor
$F_{\gamma\to\pi^+\pi^0\pi^-}=F^{3\pi}$ both taken at vanishing momenta of
mesons. Our approach is based on a recently proposed covariant diagonalization
of $\pi a_1$-mixing within a standard effective QCD-inspired meson Lagrangian
obtained in the framework of the Nambu-Jona-Lasinio model. We show that the two
surface terms appearing in the calculation of the anomalous triangle quark
diagrams or AVV- and AAA-type amplitudes are uniquely fixed by this theorem. As
a result, both form factors $F^\pi$ and $F^{3\pi}$ are not affected by the $\pi
a_1$-mixing, but the concept of vector meson dominance (VMD) fails for
$\gamma\to\pi^+\pi^0\pi^-$.
