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[tlaplus] Q about the existential quantifier

hello, the Specifying Systems book says that a regular existentially quantified variable functions as a constant. And from the definition this does seem to be the case
    \sigma |= (\E x: F) == \E x: (\sigma |= F)
where \sigma is a trace. However, if I have 
    [] (\E x: F)
its now possible for each state of \sigma to have a different value of x and satisfy this formula. But this now seems closer to a temporal existential. So would it be equivalent to 
    \EE x: [] F

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