We discuss theoretically phase transitions in frustrated antiferromagnets with biaxial anisotropy or dipolar forces in magnetic field applied along the easy axis at T = 0. There are well-known sequences of phase transitions on the field increasing: the conventional spin-flop transition and the flop of the spiral plane at strong and weak easy-axis anisotropy, respectively. We argue that much less studied scenarios can appear at moderate anisotropy in which the magnetic field induces transitions of the first order from the collinear state to phases with spiral orderings. Critical fields of these transitions are derived in the mean-field approximation and the necessary conditions are found for the realization of these scenarios. We show that one of the considered sequences of phase transitions was found in multiferroic MnWO4 both experimentally and numerically (in a relevant model) and our theory reproduces quantitatively the numerical findings.