Just as Yom Tov has the power to cut short a shiva, it has the power to cut short the Shloshim, provided that the relative began to observe the Shloshim, even momentarily, before Yom Tov.
Accordingly, if the relative sat through the complete shiva, and the seventh day of the shiva was Erev Yom Tov, he may shave and cut his hair in the honor of Yom Tov. In this situation, unless the deceased is a parent, the Shloshim will not be resumed after Yom Tov. This is because, at least for a moment in time on Erev Yom Tov, between getting up from the shiva and shaving, the relative observed the Shloshim. Although Yom Tov does not have the power to cancel the Shloshim that was not begun before Yom Tov, it does have the power to shorten it. This is because the days of Yom Tov are considered part of the Shloshim.
Accordingly, if the deceased died on Erev Pesach and a moment of shiva was observed before Yom Tov, only 15 days of Shloshim will be observed after Pesach. This is because the moment of shiva observed before Yom Tov, for which one gets 7 days’ credit, plus the 8 days of Yom Tov itself, are deducted from the Shloshim.
If the deceased died on Erev Shavuot and a moment of shiva was observed before Yom Tov, only 15 days of Shloshim will be observed after Shavuot.
If the deceased died on Erev Rosh Hashanah and a moment of shiva was observed before Yom Tov, only seven days’ Shloshim will be observed after Rosh Hashanah. This is because the arrival of Yom Kippur cancels the remainder of the Shloshim observed between Rosh Hashanah and Yom Kippur. Accordingly, in such a situation, the relative may shave on Erev Yom Kippur.
If the deceased died on Erev Sukkot and a moment of shiva was observed before Yom Tov, only eight days of Shloshim will be observed after Sukkot. This is because the moment of shiva observed before Yom Tov for which one gets 7 days’ credit, plus the 8 days of Sukkot, plus Shemini Atzeret, a Festival all its own, for which one gets 7 days’ credit, plus 1 day Simchat Torah are deducted from the Shloshim. The reason why Shemini Atzeret, does not cancel the Shloshim but only shortens it is because no Shloshim were observed before Yom Tov. You must experience the mourning of the Shloshim, even for a moment. If you did not, Yom Tov cannot cancel it.
If mourning is incompatible with Yom Tov, why is it not incompatible with Shabbat? Yom Tov is about outward manifestation of joy, simcha, whereas Shabbat is about inward serenity, oneg. On Shabbat we celebrate creation. Both life and death are part of creation. Oneg Shabbat is the attempt to come to terms with the way God created the universe.