You do understand that the FX lenses work fine on DX cameras, right?
Yes but then there are the differences in focal length and depth of focus. I'd rather be able to switch to a second camera and not have to go with a different lens to compensate for that difference.
There are no differences in focal length between DX and FX using the same lens. As for DoF, it's subjective on how you define the comparisons.
Depth of focus is greater with the smaller DX sensor than with the larger FX sensor (assuming things like f stop, distance to subject, and final image size are the same).
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Not to continue beating an already dead horse but the DoF is the same under the conditions you have listed. If, however, you move the FX so as to equal the same coverage as the DX will then the DoF will be different. But keeping all things constant as you have described DoF is the same. The reason is that DoF is an optical property of the lens and has nothing to do with the sensor.
Sorry, I meant to say subject distance in terms of providing the same FOV, then DOF will be greater with the DX. However, if distance to subject, f stop, focal length, and the final images are the same size, the DX will have less DOF.
Also, if you change sensor size, you would need a different angle of light to fill the sensor, and therefore a different focal length. DOF ratio is in direct proportion to format size when using the same focal length, same subject distance, and the same f-stop, and inversely proportional to format size when focal length is adjusted at same subject distances to give the same size final image. So DOF is intimately associated with format size, because it is derived from not only what the light is being projected FROM, but also where the light is being projected TO.