Let us have two random variables $A$ and $B$ representing lifetimes of two elements of a system, where $A$ has cdf $F_A(x)$, $A \sim Exp(\lambda_1 + \lambda_{12})$ and $B$ has cdf $F_B(y)$, $B \sim Exp(\lambda_2 + \lambda_{12})$, with joint cdf $H(x,y)$ .
Marshall-Olkin copula is defined as survival copula
$$\bar{H}(x,y) = C_{\theta_A, \theta_B}(u,v) = \min (u^{1-\theta_A}v, uv^{1-\theta_B})$$
where $$\theta_A = \frac{\lambda_{12}}{\lambda_1+\lambda_{12}} \text{ and } \theta_B = \frac{\lambda_{12}}{\lambda_2+\lambda_{12}}$$
$$u = \bar{F_A}(x) = 1 - F_A(x) \text{ and }v = \bar{F_B}(y) = 1- F_B(y)$$
What is the formula for non-survival Marshall-Olkin copula that would have $u = F_A(x)$ and $v=F_B(y)$?