# Can an Arcane Trickster/Wizard multiclass write down AT spells into its spellbook? [duplicate]

It is said, AT are half-wizards who know their spells so well they don't need a spellbook to cast them. Wizards can read and write down spells, make Scrolls or copy them.
Can these perfect entities write their own, well-known spells into their spellbook by heart?

My example:

I have a 9AT/1W character.
My AT knows the Sleep spell and my spellbook contains Charm Person.
After I leveled up as AT, I have an option to change one of my spells to another one which is illusion or enchantment.

Can I change Sleep to Charm Person, meanwhile I write Sleep in my spellbook?

In this case, I don't lose any of my spells but I can prepare other, more useful spells as a Wizard and Charm Person will be always "prepared".
I could do the same if I changed the spells anyway and then bought a Sleep scroll somewhere but this is more time-consuming and illogical.