The jihadists also confirmed the death of their former spokesman Abu Hassan al-Muhajir in a separate operation.
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However, the extremists warned America not to ‘rejoice’ in the death of two senior ISIS figures.
Warning that ISIS was not limited to the Middle East, the terrorists vowed that their extremist ‘mission’ would continue.
Terror expert Rita Katz said the message showed that ISIS’s online operations were ‘alive and well’ despite Baghdadi’s death.
The appointment of al-Qurayshi as the new ISIS leader contradicts rumours that Baghdadi’s former security henchman Abdullah Qardash was set to take the top job.
Baghdadi died after fleeing into a network of underground bunkers and tunnels that snaked through the compound in Syria, Trump revealed on Sunday.
‘He reached the end of the tunnel, as our dogs chased him down,’ Trump said. ‘He ignited his vest, killing himself and the three children.’
Trump also declared two days ago that Baghdadi’s ‘number one replacement’ had been ‘terminated’, apparently referring to spokesman Muhajir.
Muhajir was killed on Sunday in a joint raid between Kurdish-led and U.S. forces in northern Syria, a State Department official confirmed on Monday.
Russia had previously voiced doubts about whether Baghdadi was really dead after numerous false rumours of his death over the years.
Baghdadi had been detained in Iraq after the U.S.-led invasion which toppled Saddam Hussein in 2003.
The operation against Baghdadi has already prompted fears of revenge attacks by followers of the terrorist leader.
However, Baghdadi was widely seen as a figurehead rather than a day-to-day leader and his followers are still capable of mounting attacks.
Security has been tightened at ISIS prisons in northern Syria where Kurdish guards were said to be on ‘high alert’.
Baghdadi had urged his followers to break ISIS fanatics out of prison in the last weeks of his life.