Why is $AF \phi \lor \varphi$ not equivalent to $AF \phi \lor AF \varphi$? Can someone explain to me (perhaps with an example) why $AF (\phi \lor \varphi)$ is not equivalent to $AF \phi \lor AF \varphi$. This seems counter-intuitive, because in any path where $\phi$ (or $\varphi$) is true, $\phi \lor \varphi$ is true as well. Or is my reasoning wrong?

Thanks
You also need to argue the other way around. &quot;Each path has either $\psi$ or $\varphi$&quot; is not the same as &quot;either each path has $\psi$ or each path has $\varphi$&quot;.