Okay so a little wonky. I play Sylvan Library. My opponent plays Chains of Methostopolies. What happens at my draw phase?
First, you draw a card for your turn. Nothing special happens. Be sure to keep the cards you’ve drawn this turn separate from the other cards in your hand until after Sylvan Library’s trigger finishes resolving.
Second, you choose if you want to draw the two extra cards when Sylvan Library’s trigger starts to resolve. I recommend against this. Let’s say you choose to draw the cards, though, since otherwise this answer isn’t very interesting.
So, now you’ll draw the first card for Sylvan Library, but Chains makes you discard a card first. We know you have a card because you drew for your turn, so you’ll discard a card, then draw a card. Then you repeat this process for the second card.
Finally, you’ll look at how many cards you have left that you’ve drawn this turn. You’ll have at least one from that last draw. If you just have the one, you have to pay 4 life or put it back on top of your library. If you have two or three, then choose two of them and you have to pay 4 life for each one you want to keep or put them back on top.
