my $statement = 'SELECT a1.ID FROM a1, a1_b1, b1 WHERE b1.id = ? AND a
+1 = ?';
my $sth = $dbh->prepare($statement) or die $dbh->errstr();
for my $a1(@array1){
    for my $b1 (@array2){
        $sth->execute($b1,$a1) or die $dbh->errstr();
        # ... other stuff ...
    }
}
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