User blog comment:Deedlit11/Ordinal notations II: Up to the Bachmann-Howard ordinal/@comment-5150073-20130628190402/@comment-5529393-20130628195020

That statement is sometimes true, except when \(\psi(\alpha)\) stabilizes. For example, it is true when \(0 \le \alpha < \varphi(2, 0)\), but not true when \(\varphi(2, 0) \le \alpha < \Omega\). In the latter case \(\psi(\alpha + 1) = \psi(\alpha)\).

I'm not sure what you mean by "all rules for theta function remain the same." The same as what?