That is not true, you can easily check that the following formula is valid using the LTL prover:

G(!a&!b) -> !(FG(a&!b) nor G(a&!b) & G!b) & !F G(a&!b)

Hence, both formulas are false. It is clear that G!b holds, but neither FG(a&!b) nor G(a&!b) would hold, since a does not hold always (as you wrote). But FG(a&!b) is not equivalent to G(a&!b). Moreover, F[a WU b] is equivalent to F (b | G a) which is also not true.