Given the same focusing distance, depth-of-field will not change.
That's not necessarily true. The reckoning of depth of field is based, among other things, on an arbitrary criterion we choose, the circle of confusion diameter limit (COCDL). This is the largest diameter of the blur circle resulting from imperfect focus of objects not at the ideal focus distance that we, based on some outlook, are willing to consider "acceptable".
We may use many premises to choose a value of that. One traditional premise is as a certain fraction of the frame diagonal size. The concept behind this a fixed angular size of the "blur circle" when the image itself is viewed with a consistent angular size, for example, printed to a certain size and viewed from a certain distance.
If we follow that premise, then for a camera with a "smaller format", with a certain focal length lens, a certain f-number, and a certain distance to the subject, the calculated DoF will be smaller than for the larger-format camera - not because anything has changed in the optical physics but rather because our criterion of acceptable blur is changed for the new format size.
Given the same framing, depth-of-field will be greater on the 7D because your focusing distance will be greater (due to the crop factor, you'll have to back up)
But it is interesting to see how that happens. Two factors are at work.
Firstly, if we indeed hold to the choice of a COCDL that is a fixed fraction of the fame size, then, for a "smaller" frame size, that factor of itself will decrease
the calculated DoF.
However, as you point out, to maintain the original framing, the needed distance to the subject will increase. This increases
the calculated DoF, and by a greater ratio than it is diminished by the adoption of a smaller COCDL. Thus, the net effect is indeed an increase in the DoF.