Well, if we want to give an unclunky, simplified version of the above rule, then it would be something like,

"Gluing on the common elements \$$b\$$ with the feasibility relations \$$\phi(a,b)\$$ and \$$\psi(b,c)\$$, and then gluing on the common elements \$$c\$$ with \$$\nu(c,d)\$$, is logically the same as gluing on the common elements \$$c\$$ with the feasibility relations \$$\psi(b,c)\$$ and \$$\nu(c,d)\$$, and then gluing on the common elements \$$b\$$, with the feasiblity relation \$$\phi(a,b)\$$."