The omnibus test comes to be significant mostly if at least one of the independent variables is significant. Which means that any other variable may enter the model, under the model assumption of non-colinearity between independent variables, while the omnibus test still show significance, that is: the suggested model is fitted to the data. So significance of the omnibus F test (shown on ANOVA table) followed by model selection, which part of it is related to selection of significant independent variable which contribute to the dependant variable variation.