"Anybody / anyone" and "nobody / no one" do not mean the same thing and are not interchangeable.
- Nobody/no one mean NO persons.
- Anybody/anyone mean ANY person.
"Anybody can come to my party" would mean your party is open to any person. Everybody can come.
"Nobody can come to my party" would mean that no person can come to your party. There will be no guests at all. Not much of a party, really.
Your example does not work with "anybody", because "anybody" is a non-specific person. You can't say "anyone was there". Unlike the party invitation where "anybody can come", When speaking in the past tense you can be specific and say who, or who wasn't actually there. If there were no persons there then you should say "nobody was there".
You can use "anybody/anyone" in a negative way to mean nobody, for example:
I don't think anybody saw me.
I didn't see anyone.