That is not true actually. There are a number of processes involved. One is the cooling of the stratosphere as a result of the reduction in ozone content. Then the possible modifications in stratospheric dynamics due to the change in ozone distribution. For example it is argued that the polar vortex over the Antarctic, which confines the ozone 'hole,' takes longer to break in the spring due to additional cooling resulting from major ozone depletion in the region.
Ozone is a powerful greenhouse gas.
The change in stratospheric dynamics is now thought to affect the tropospheric dynamics because of the resulting changes in the dynamics. The results I am mentioning, by Bitanja et al. J. of Climate 1997 consider the net effect of cooling resulting from stratospheric ozone depletion and the heating resulting from the CFCs. The balance of these two processes results in the figure I mention. Furthermore, recent results from the Hohenpeissenberg station in Germany do suggest that the tropopause height has changed.