I see. I didn't realize that part. Try this:
=IFERROR(IF(AND(MATCH($D13,'Master Entry List '!$A$2:$A$20,0)=MATCH($E13,'Master Entry List '!$B$2:$B$20,0),MATCH($D13,'Master Entry List '!$A$2:$A$20,0)=MATCH($F13,'Master Entry List '!$C$2:$C$20,0),INDEX('Master Entry List '!$A$2:$M$20,MATCH($D13,'Master Entry List '!$A$2:$A$20,0),13=1)),"Yes",""),"")
What it does, is finds the row number within the range A2:A20 of the first name and compares it to the row number of the last name, and compares the row number of the first name to the row number of the name of the horse, and checks to see if there is a 1 in the M column at the same row as the first name. If any of those is false, the cell will be blank. If all 3 of those are true, it will return 'Yes.' The "IFERROR" part will prevent errors from showing up when you fill down to a row where there are no entry names.
One last addition: the last row of ranges used ($A2:$A20, for instance) will need to be adjusted if there are more than 20 rows of data. If there are 50 entrants, the column A range will be $A2:$A51, as long as there are no skipped rows.
I just checked something and it will only work if the first names are all unique. So, if there is more than one "Julie," it doesn't work. This will work until you get to that point. :/
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