[semi-extrinsic]

The key clarification is in one of the comments: if you want to treat partial derivatives like fractions, you need to carry the "constant with respect to foo" modifier along with both nominator and denominator.

Once you do that, it's clear that you can't cancel "dx at constant z" with "dx at constant y" etc. And then the remaining logic works out nicely (see thermodynamics for a perfect application of this).

[yccs27]

This is the crucial insight. The last proof in TFA writes it in differential form notation, where "dx at constant y" is dx∧dy = -dy∧dx.

[quibono]

I think the video linked in the post does this, except for the 2-dimensional case.