MySQL Cannot drop index needed in a foreign key constraint

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I need to ALTER my existing database to add a column. Consequently I also want to update the UNIQUE field to encompass that new column. I’m trying to remove the current index but keep getting the error MySQL Cannot drop index needed in a foreign key constraint

CREATE TABLE mytable_a (
ID          TINYINT NOT NULL AUTO_INCREMENT PRIMARY KEY,
Name        VARCHAR(255) NOT NULL,
UNIQUE(Name)
) ENGINE=InnoDB;

CREATE TABLE mytable_b (
ID          TINYINT NOT NULL AUTO_INCREMENT PRIMARY KEY,
Name        VARCHAR(255) NOT NULL,
UNIQUE(Name)
) ENGINE=InnoDB;

CREATE TABLE mytable_c (
ID          TINYINT NOT NULL AUTO_INCREMENT PRIMARY KEY,
Name        VARCHAR(255) NOT NULL,
UNIQUE(Name)
) ENGINE=InnoDB;


CREATE TABLE `mytable` (
  `ID` int(11) NOT NULL AUTO_INCREMENT,
  `AID` tinyint(5) NOT NULL,
  `BID` tinyint(5) NOT NULL,
  `CID` tinyint(5) NOT NULL,
  PRIMARY KEY (`ID`),
  UNIQUE KEY `AID` (`AID`,`BID`,`CID`),
  KEY `BID` (`BID`),
  KEY `CID` (`CID`),
  CONSTRAINT `mytable_ibfk_1` FOREIGN KEY (`AID`) REFERENCES `mytable_a` (`ID`) ON DELETE CASCADE,
  CONSTRAINT `mytable_ibfk_2` FOREIGN KEY (`BID`) REFERENCES `mytable_b` (`ID`) ON DELETE CASCADE,
  CONSTRAINT `mytable_ibfk_3` FOREIGN KEY (`CID`) REFERENCES `mytable_c` (`ID`) ON DELETE CASCADE
) ENGINE=InnoDB;




mysql> ALTER TABLE mytable DROP INDEX AID;
ERROR 1553 (HY000): Cannot drop index 'AID': needed in a foreign key constraint

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