It's slightly more intuitive to think about this problem in terms of conditional probabilities.
We want to understand why P(F | G) = P(F). There are exactly 6 pairs of dice values whose sum is 7: {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}. That is for each possible outcome of the first roll, there is exactly one (unique) outcome of the second roll which would yield a sum of 7. In other words, given the value of the first die, the probability that the sum of the dice roll is 7 is 1/6.
P(E|G) != P(E) because there are values of the first roll which make it impossible to have a sum of 6 - in particular, if the value of the first roll is 6. Therefore, knowledge of the value of the first die changes the probability that the sum of the two dice is 7.