Vanilla Minecraft does not have an option to wear an elytra and a chestplate simultaneously. You either have to sacrifice protection for flight, or sacrifice flight ability for protection. Most players find themselves switching between their chestplate and elytra constantly. Life would be so much more easier if there was an option to wear these items together. And this is where the Curious Elytra mod helps you. This mod adds an extra slot that you can use to equip an elytra. This means your chestplate slot will remain empty and you can equip a chestplate there for your protection. The ability to wear these items together will eliminate the need to swap between these items constanly and will make your gameplay expereience a lot more better.

This is a Forge mod and download links are given below.

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Curious Elytra Mod