How Do Birds Get Vitamin D3?

In the book Beaks, Bones, and Bird Songs by Roger Lederer, Lederer says that, when pruning, birds put a chemical on their feathers that becomes Vitamin D.

Very little sunlight reaches their skin through their feathers, so they need another mechanism to gain Vitamin D.

They put a chemical on their feathers while pruning; the chemical gets turned into Vitamin D by sunlight; the birds ingest the Vitamin D next time they prune.

Wow!