Linear Temporal Logic, Idempotent law
Also by definition of $G$, some formula holds in every state.


Well, $\sf true$ holds in every state. I'm unsure about why you mention $G$ here.


  So if globally $\phi$ holds until we see $\psi$ does it make every state after it true?


A state can not be true or false, like 42 can not be true or false. This is meaningless.


  is it correct to say that also $G(\phi U \psi)U\phi \equiv G(\phi U \psi)  $


Take $\phi = \sf false$ and $\psi = \sf true$. Then, $G(\phi U \psi)U\phi$ is equivalent to $\sf false$, and $G(\phi U \psi)$ is equivalent to $\sf true$.