Does Bayes theorem hold for variance?

Question

$\DeclareMathOperator{\Var}{Var}$ I am trying to find $\Var[A\mid B]$ of a random walk when $A < B$. I was wondering whether i can apply the Bayes theorem which says that

$P[A\mid B]$ = $\dfrac{P[B\mid A]P[A]}{P[B]}$

to the variance such that

$\Var[A\mid B]$ = $\dfrac{\Var[B\mid A]\Var[A]}{\Var[B]}$