  We prove the Bianchi permutability (existence of superposition principle) of
B\"acklund transformations for asymmetric quad-equations. Such equations and
there B\"acklund transformations form 3D consistent systems of a priori
different equations. We perform this proof by using 4D consistent systems of
quad-equations, the structural insights through biquadratics patterns and the
consideration of super-consistent eight-tuples of quad-equations on decorated
cubes.
