Checking whether a Mixed Strategy Nash Equilibrium is Evolutionary Stable

Question

Taking the classic Hawk-Dove game

I know that there are 3 mixed strategy Nash Equilibria, (H,D), (D,H), and the mixed strategy ((7/12, 5/12), (7/12,5/12)), where those are probabilities of playing H and D respectively per player.

We just started ESS (Evolutionary Stable Strategies), and from my understanding, an ESS A must be a symmetric Nash Equilibrium where the payout of player 1 choosing A and 2 choosing B:

u(A, B) > u(B, B) for any B a best response of A.

My logic is as follows:

1. The pure strategies are not ESS, because they are not symmetric NE.
2. The mixed strategy must be, because a theorem says that any symmetric game must have at least 1 ESS. BUT, I would like to prove that this strategy is indeed an ESS, but I'm stuck as to how to go about doing it? How do we computationally prove that u(A, B) > u(B, B) for any B a best response of the mixed strategy?

Thank you, and I hope my work so far is clear!