Let $X=(X_1,X_2)$ and $Y=(Y_1,Y_2)$ be non-negative absolutely continuous random vector and if $\phi(X_j)=Y_j$, $j=1,2$, are one-one transformation then $$H[Y;\phi(t_1),\phi(t_2)]=H(X;t_1,t_2)-E[\log J(X_1,X_2)|X_1>t_1,X_2>t_2]~~~~~~~~~~~~(1)$$ where $J(y_1,y_2)=\left|\frac{\partial\phi_1^{-1}(y_1)}{\partial y_1}\times\frac{\partial \phi_2^{-1}(y_2)}{\partial y_2}\right|$, and $H(X;t_1,t_2)=-\int_{t_2}^\infty\int_{t=1}^\infty\frac{f(x_1,x_2)}{\bar{F}(t_1,t_2)}\log\left(\frac{f(x_1,x_2)}{\bar{F}(t_1,t_2)}\right)dx_1dx_2$, $\bar{F}(x_1,x_2)=Pr(X_1>x_1,X_2>x_2).$
RHS of the equation(1) can be written as
$$H(X;t_1,t_2)-E[\log J(X_1,X_2)|X_1>t_1,X_2>t_2]=-\int_{t_2}^\infty\int_{t=1}^\infty\frac{f(x_1,x_2)}{\bar{F}(t_1,t_2)}\log\left(\frac{f(x_1,x_2)}{\bar{F}(t_1,t_2)}\right)dx_1dx_2-\int_{t_2}^\infty\int_{t=1}^\infty\frac{f(x_1,x_2)J(x_1,x_2)}{\bar{F}(t_1,t_2)}dx_1dx_2.$$
I want to prove this. I am new in statistics field. I don't know some specific theorem that will be use here in order to use transformation.
I can write
\begin{align} H[Y;\phi(t_1),\phi(t_2)]&=-\int_{\phi_2(t_2)}^\infty\int_{\phi_1(t_1)}^\infty\frac{f(y_1,y_2)}{\bar{F}(\phi_1(t_1),\phi_2(t_2)}\log\left(\frac{f(y_1,y_2)}{\bar{F}(\phi_1(t_1),\phi_2(t_2))}\right)dy_1dy_2 \end{align}
\begin{align}{\bar{F}(\phi_1(t_1),\phi_2(t_2))}&=Pr(Y_1>\phi_1(t_1),Y_2>\phi_2(t_2))\\&=Pr(\phi_1(X_1)>Pr(Y_1\phi_1(t_1),Y_2>\phi_2(t_2))\\ &=Pr(X_1>t_1,X_2>t_2). \end{align} I don't know how to proceed further.