Does this notation for partial derivatives work?

[Gerenuk]

I noticed that for partial calculus equations I can use Jacobians

\left(\frac{\partial A}{\partial B}\right)_C=\frac{\partial(A,C)}{\partial(A,B)}

You immediately get the triple product rule
(http://en.wikipedia.org/wiki/Triple_product_rule)

\frac{\partial(A,C)}{\partial(A,B)}=\frac{\partial (A,C)}{\partial(A,B)}\frac{\partial(B,C)}{\partial (B,C)}=-\frac{\partial(A,C)}{\partial(B,C)}\frac{\partial( B,C)}{\partial(B,A)}

and of course the other rule

\frac{\partial(A,C)}{\partial(A,B)}=\frac{\partial (A,C)}{\partial(A,B)}\frac{\partial(A,D)}{\partial (A,D)}=\frac{\frac{\partial(A,C)}{\partial(A,D)}}{ \frac{\partial(A,B)}{\partial(A,D)}}


Now I was surprised to see that even the pseudo-equation for the chain rule

\partial(A,B)=\frac{\partial(A,D)\partial(E,B)-\partial(A,E)\partial(D,B)}{\partial(E,D)}

works well even if I treat all terms as a real variable.

My question is: Does this algebra for partial derivatives always work despite crazy manipulations and cancellations?