Is the inverse of Armstrong's Axioms valid as well?
No, that is not true. 

Armstrong's axioms are a sound and complete  axiomatization of the logical implication for functional dependencies.

Here is a relation, where the functional dependency XZ → YZ holds:

X  Y  Z
--------
x1 y1 z1
x2 y1 z1
x1 y2 z2
x2 y2 z2


But the functional dependency X → Y does not hold, as the tuples x1 y1 z1 and x1 y2 z2 show. Because of the soundness of Armstrong's axioms, which means that only valid functional dependencies can be derived, the "inverse" is not true. So we have constructed a counterexample.



If this "inverse" law would hold, then we can show that X → Y for arbitrary attribute sets X and Y:

Y → Y      (reflexivity)
XY → XY    (augmentation)
XY → Y     (reflexivity)
XY → YY    (idempodency of union-operator: Y=YY)
X → Y      ("inverse")