I'm reading the Multivalued dependency page of Wikipedia. There are some properties given for multivalued dependencies. But the following property is not given:
if A →→ BC then A →→ B and A →→ C
(→→ indicates a multivalued dependency.)
Is this valid for multivalued dependencies?
The Armstrong’s Axioms hold only for Functional Dependencies, not for Multivalued Dependencies (in fact in the wiki page you can see that, even if they have sometimes the same name, the properties are different).
In particular, the decomposition rule (the property in your question) does not hold form Multivalued Dependencies.
This is discussed in several database books. For instance, there is an example about this in the book of Garcia-Molina, “Database Systems: The Complete Book.” 2nd ed. Upper Saddle River, N.J, Pearson Prentice Hall, 2009, at page 109. Here, a relation about Hollywood stars is presented, with their (multiple) addresses and (multiple) titles and years of their movies. So the relation has attributes:
(name, street, city, title, year)
with the MVDs:
name →→ street, city (a star can have multiple addresses)
name →→ title, year (a star can have multiple movies)
From these dependencies you cannot infer, for instance, that:
name →→ street
name →→ city
since you loose the relation between the street and the city of an address of a star.