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1.
Much like an antipope, this is a man who has made an improper claim to the Preisdency of the United States.

The current occupant of the White House.
George W. Bush is the Antipresident.
by MrClean February 06, 2006
 
2.
Any figure in a Republic who claims to be the President of that specific nation despite not being elected by the popular vote OR claiming to be the President even whilst an elected President is in office.
Jefferson Davis, as head of the Confederate States of America, could be considered an 'Anti-President'.
by Tyburn January 15, 2013