"Levant" is used as a translation of part of the Arabic name, ISIL (ISIS); an old term for the eastern Mediterranean area now Syria, Lebanon, Israel and Jordan. This is not what is meant by the leader and self-proclaimed caliph, Abu Bakr al-Baghdadi.

More elements of the rebels fighting the Syrian government seem to be falling to ISIL. Most of the villages along the Euphrates River are under ISIL control.
Associated press, July 3.