Question

I've seen that Anaxagoras (500 BCE–428 BCE) was the first to propose that the Moon reflects the Sun's light, but when was that assertion first scientifically proven, and who did it?

Answer 1

They are are one and the same! His assertion was based on scientific observation of the phenomenon.

Explanation:

The scientific "proof" was the observation of an eclipse, which inspired his description of the phenomenon. When the sun's light was blocked from the moon, its shine diminished. This could then be further substantiated by the explanation of the moon light phases.

Answer 2

Leonardo Da Vinci and Johannes Kepler

Explanation:

In Leonardo Da Vinci's unpublished works, he proved the concept of planet shine.
Here is a diagram from his Codex Leicester book.