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If you have the Apple Card already added to Wallet, you don't need to do anything for a new iPhone. Just sign in with iCloud.

If you have a new iPhone and have not paired the metal credit card, if you have the original packaging all you need to do is bring them close, like AirPods.

Here's an example:
(0:45 seconds in)

If you don't have the packaging you can do it manually through the Wallet app.
 
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