In Projective Geometry, Fano's Axiom says:
The three diagonal points of a complete quadrangle are never collinear.
I would like to prove this from more basic Axioms within three-dimensional Projective Space. The theorem of Desargues is non-trivial in plane geometry, but can be proven from basic axioms within three-dimensional geometry; could the same be true for Fano's Axiom? If not, are there nice (equivalent) alternatives?
Edit: by “more basic” I mean intuitively more basic, which makes it a somewhat subjective question of course.