Kudo transgression theorem

Steenrod operations preserve transgressive pairs. With mod-two coefficients, if $(x,y)$ is transgressive, then so is $({\mathrm{Sq}}^{j}x,{\mathrm{Sq}}^{j}y)$ whenever the square is defined by the instability range. The corresponding statement holds for odd-primary reduced powers and the Bockstein, with the appropriate page shift.