Well, if everyone went vegan then sure, farmers would go out of business - but couldn't those same animals just fend for themselves and live in the wild? What makes you so sure they'd go extinct?
I actually know a few points that support the op's argument, but if he's too lazy to support his own topic I'm not going to do it for him.
Then I will do it myself.
1) There is plenty of evidence to support the idea that if everybody went vegan, farmer would no longer have any financial motivation to raise animals in the first place. How will farmer even be able to afford the to purchase the food if he has no source of income. If you want proof of this theory from farmers directly, I have a video of a farmer claiming that there would be no farm animals in a vegan world. He says to a bunch of angry vegan protesters "do you realise that if people had your way, do you think that will be better for the animals? The will be no animals, why would there be any animals, we can produce animals for nothing can we", one of the vegan protestors asks him why there would be no animals, and the farmer responds with "well because, we only produce them for food". Since this is a farmer saying this, nobody else can refute unless they understand animal farming better than an animal farmer can. Well, here is the video, go to 00:20 and you will see the farmer saying this: https://www.youtube.com/watch?v=RDukZKYuQwE&t=364s
2) If we we release domestic animals into the wild, they would likely have a hard time surviving because they domesticated animals have lost most of their survival instincts. Domestic sheep are way too friendly and docile, they have no idea how to spot or outrun predators since they have been living under human protection for far too long. Try putting a domestic sheep in the middle of the African wilderness, and see how long it survives against all the lions and hyenas. Why wouldn't they go extinct?