The BYC rule applies to a specific circumstance. If a player is being signed-and-traded via Early Bird or Bird rights by a team above the salary cap, gets a raise of at least 20%, and his salary is worth more than the minimum, his cap figure for salary-matching purposes will be affected. For the team acquiring him, his full salary would apply in a trade. For the team trading him, he would count for his previous salary or 50% of his new salary, whichever is greater.
The Bulls will have the cap space however to make this work.
Chicago in: Tomas Satoransky (signed at 4 years $44 million)
Washington in: Kris Dunn and Antonio Blakeney (combined $5.5 M)
I really like Satoransky's size, playmaking and shooting either alongside the Bulls current starters or off the bench (depending on other free agent plans and the draft).
For Washington - they're in a bit of a odd situation with Wall's injury and Satoransky's desire to be seen as a point guard:
“I’m a point guard. I’m definitely a point guard. I will never be agreeing with someone that tells me otherwise,” Satoransky said. “I know how I feel here.”
Source: https://www.nbcsports.com/washington/wizards/wizards-pass-first-guard-tomas-satoransky-gets-his-shot-defining-his-role-free-agency
Dunn and Blakeney (the later really just being there for salary) both only have a single season remaining on their current contracts, giving the Wizards a point guard during Wall's recovery without committing long term money to a position where they already have a lot of money committed (Wall).
So it helps Chicago sign Satoransky with only half the money coming for their cap space, and helps Washington find a short-term stop gap during Walls recovery that they could decide they like and have matching rights after the season.