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=and(countif($A$1:$B$5,A1)=1,countif($D$1:$D$5,A1)=1)
This does seem to check for "No formatting if there are duplicates in A1:B5" but also seems to check for duplicates in D1:D5.
Why isn't A5 highlighted here:?
I would put forward:
=AND(COUNTIF($A$1:$B$5,A1)=1,COUNTIF($D$1:$D$5,A1)>0)
which yields: