In a thermal physics book by Blundell and Blundell, the following (paraphrased) theorem is proved:
If $x=x(y,z)$, $y=y(x,z)$ and $z=z(x,y)$, then $$\left(\frac{\partial x}{\partial y}\right)_z = -\left(\frac{\partial x}{\partial z}\right)_y \left(\frac{\partial z}{\partial y}\right)_x$$ where subscripts indicate the variable being held constant in the derivative.
To show this, the authors simply wrote out the total differential for each of $\text{d}x$ and $\text{d}z$ and then substituted the one into the other, yielding
$$\text{d}x = \left(\frac{\partial x}{\partial z}\right)_y \left(\frac{\partial z}{\partial x}\right)_y \text{d}x + \left[\left(\frac{\partial x}{\partial y}\right)_z + \left(\frac{\partial x}{\partial z}\right)_y \left(\frac{\partial z}{\partial y}\right)_x \right]\text{d}y.$$
Next, they concluded that the corresponding coefficients of $\text{d}y$ must be the same, thus proving the theorem (since the coefficient of $\text{d}y$ on the LHS is implicitly $0$).
Ordinarily, I would be fine with this reasoning, assuming $x$ and $y$ are independent variables. However, since $x$, $y$ and $z$ were assumed to all be functions of each other, I'm not so convinced.
Can anyone provide further justification for this reasoning? I don't necessarily require a formal proof, a clear explanation of why this reasoning is justified would hopefully suffice.
We can first analyze a expression, important in the proof you've cited: the relation between the derivative of a function and the derivative of its inverse. Overcoming the difficulties with this expression, belonging to the easier case of one single variable, was for me important to understand similar expressions in two or more variables.
We are combining in a single expression $y'(x)$ and $x'(y)$, being $x$ and $y$, as you said, functions of each other! We can use the Leibniz notation "freely" to see quickly the relation between them, though it's not rigorous: $\dfrac{\mathbb d x}{\mathbb d y}=\dfrac{1}{\dfrac{\mathbb d y}{\mathbb d x}}$. If we want some rigor we have to write a full sentence: $g'(x)=1/f'(g(x))$ for $f$ and $g$ such that $f(g(x))=x$, so is, the expressions have to bear the dependence. A point in between is to use the Leibniz notation and think, so say, geometrically: the first formula says that in a determinate point $(x,y)$ the slope of the tangent line considering $y$ as function of $x$ is the reciprocal of the slope considering $x$ as function of $y$. If you keep in mind the idea of the "absolute" graph of the function, better.
So, "think geometrically": we have a point $(x,y,z)$ and we have at that point six quantities to work with: $\dfrac{\partial x}{\partial y}$; $\dfrac{\partial x}{\partial z}$; $\dfrac{\partial y}{\partial z}$ and their reciprocals, related as their counterparts for single variable functions, e.g.: $\dfrac{\partial x}{\partial y}=\dfrac{1}{\dfrac{\partial y}{\partial x}}$. Now, the total differential: geometrically it describes a plane, tangent to some, say, absolute surface (no mind which pair of coordinates we use as free variables the surface is not a different one!): $\left(-1,\dfrac{\partial x}{\partial y},\dfrac{\partial x}{\partial z}\right)$ describes a vector perpendicular to this plane, but $\left(\dfrac{\partial z}{\partial x},\dfrac{\partial z}{\partial y},-1\right)$ too! So is, the relations have to be as easy as the geometry of the problem and it justifies the reasoning. Nevertheless if you need something more rigorous it happens as with the single variable case: you need to bear the dependence, but to me, the geometrical image is strong enough (and at the very least, rigorous too) to be by itself a very good intuition in a lot of problems.