I'm referring to a book where E and A were defined in text as follows
The formula $A \phi$ states that all the executions out of the current state satisfy property $\phi$ whereas $E\phi$ states that from the current state, there exists and execution satisfying $\phi$
And formally, they were defined as
$\sigma, i \models E\phi $ iff there exists as $\sigma'$ such that $\sigma(0)\ldots\sigma(i) = \sigma'(0)\ldots\sigma'(i)$ and $\sigma', i \models phi$ $\sigma, i \models A\phi $ iff for all $\sigma'$ such that $\sigma(0)\ldots\sigma(i) = \sigma'(0)\ldots\sigma'(i)$, we have $\sigma', i \models phi$
$\sigma(i)$ was previously defined to be the ith state of an execution $\sigma$.
Here's where my confusion lies. Doesn't $\sigma(0)\ldots\sigma(i) = \sigma'(0)\ldots\sigma'(i)$ imply that the executions $\sigma$ and $\sigma'$ have both the same start and end state, therefore $\sigma', i \models \phi$ is same as $\sigma, i \models \phi$? That seems to be wrong.