If Facebook tells you the code worked, then you're all set for logging into Facebook using Authy on the desktop.
Most services that offer two-factor authentication support use a similar system to generate security codes, though the exact method will vary by service. Try digging in the security settings of other services first.
Be warned: Many online accounts don't like it when you use more than one authenticator method for two-factor authentication, so you may have to dump text/email authentication and start using Authy exclusively on your smartphone and PC if you want to use the service for all your accounts. (Authy can sync your two-factor authentication codes across its PC and smartphone apps to make things easier.)
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