Technically yes, but the degree of benefit is small if compared to nutrient-fortified white bread in the context of a healthy diet.
Whole wheat bread tends to have a higher fiber content, and more micronutrients than standard white bread. Fortified white bread can close the gap between the two with nutrient fortification, but fiber tends to still be different.
When comparing whole wheat directly against white bread, whole wheat appears to be better. The degree of which it is better, however, is uncertain. Some sources claim little to no difference, some epidemiological studies claim much more.
Increasing dietary fiber seems to be implicated in many of the benefits, and can be achieved through increasing plant intake in general (fruits and vegetables included).
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