Rewarding with food is standard parenting practice - for example, many parents use sweet treats to encourage their child to use the potty, or to recognise good behaviour. This gives the child the following message: "I approve of you, and so you can have sweet food" . To take it a step further, from the child's point of view, it equates eating sweet food with feeling loved. EAF is all about separating food from feelings so that children eat for physiological rather than emotional reasons.
Rewarding with food fuses food and feelings.
Consider adult comfort eating(sometimes called 'emotional eating'). This is where a person eats to fill an emotional hole rather than because her body needs it. This online resource about emotional eating talks about how being rewarded with food as a child can lead to emotional eating later in life: "These emotionally-based childhood eating habits often carry over into adulthood". Sever the link between your child feeling good about herself and your child eating treat foods, by always sticking to EAF principle 3.
I have written elsewhere about using food to reward good eating - there is some fascinating research that shows that if you reward children for eating one food by giving them another food, they will like the 'reward' food more than they did originally, and like the 'hard work' food even less! Not the result you were hoping for...
These days, parenting experts talk more about using 'sanctions' than 'punishing' but, whatever language you choose, using food to convey a message of disapproval to your child is problematic - it is the other side of the coin described above. If you cancel the ice-cream that you had promised due to bad behaviour, the link between self-esteem and eating will be reinforced just at it would if you gave your child an ice-cream because of how she had behaved.