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It deopends on whether or not the index was created when the constraint was created.
eg if you create a unique index on one or more column and then create a unique constraint on those columns, then the constraint will use the existing unique index.
in this case, when you drop the constraint the index will remain.
However if you do not have a unique index on your column(s) when you created the constraint, then when you create the constraint a unique index is also created.
In this case, when you drop your constraint, the index will also be dropped.