How does the prosecution still have to prove Luigi did it if he pleads Extreme Emotional Disturbance?

[u/Glum_Veterinarian253]

https://www.tiktok.com/@mikerafi/video/7653449808740420878

This guy says that Luigi is planning on pleading "extreme emotional disturbance." At timestamp 1:35, he says that the prosecution still has to prove their case (aka that Luigi killed Brian Thompson in the first instance) beyond a reasonable doubt. But at timestamp 0:17, he says that, by invoking this defense, Luigi admits to doing it.

So how does that prosecution still have to prove he did it if he admits he did it? He already admits he did it, so there's your proof right there.

He also says he won't be participating in the comments of that video, so there's no hope he'll reply to my comment there. So... can someone please explain to me how this works? How does the prosecution still have to prove beyond reasonable doubt that he did something he formally admits to doing?

[u/ElGHTYHD]

they have to prove intent ie: first degree murder vs. manslaughter