Using ifelse changes output it is not supposed to change

ID <- 1:10
YOB <- c("2000", "2001", "2001", "2002", "2002", "2004", "2005", "2007", "2008", "2008")
born_before <- c("FALSE", "FALSE", "TRUE", "NA", "FALSE", "NA", "TRUE", "TRUE", "FALSE", "NA") 
df <- data.frame(ID, YOB, born_before)

I am trying to use an ifelse statement to change year of birth (YOB) to blanks if an individual is born before either of its parents (if born_before is true). If born_before is NA, I would also like to change YOB to a blanks. However, with this piece of code

df$YOB <- with(df, ifelse(born_before == TRUE, YOB == "", YOB))

my YOB gets changed completely to different numbers. The resulting dataframe looks like this

ID	YOB	born_before
1	1	FALSE
2	2	FALSE
3	0	TRUE
4	3	NA
5	3	FALSE
6	4	NA
7	0	TRUE
8	0	TRUE
9	7	FALSE
10	7	NA

when it should like this one

ID	YOB	born_before
1	2000	FALSE
2	2001	FALSE
3		TRUE
4		NA
5	2002	FALSE
6		NA
7		TRUE
8		TRUE
9	2008	FALSE
10		NA

Any ideas what is going wrong? I am a novice R user (have dabbled a bit) and would like to learn. Thanks in advance!