}}</ref> with the caveat that the axiom and rules of inference are all [[Schema (logic)|schemata]], meaning that the ''X'', ''Y'' and ''Z'' range over all ground terms (attribute sets).<ref name="Vardi"/>
FromBy theseapplying rulesaugmentation and transitivity, weone can derive thesetwo secondaryadditional rules:<ref name="SilberschatzKorth2010a"/>
* '''UnionPseudotransitivity''': If ''X'' → ''Y'' and ''XYW'' → ''Z'', then ''XXW'' → ''YZZ''<ref name="SilberschatzKorth2010a"/>
* '''Decomposition''': If ''X'' → ''YZ'', then ''X'' → ''Y'' and ''X'' → ''Z''▼
* '''Pseudotransitivity''': If ''X'' → ''Y'' and ''WY'' → ''Z'', then ''WX'' → ''Z''
''X''Another →consequence of Armstrong''YZ'',s holdsaxiom [[iff]]is ''X'' → ''Y''union and decomposition''X'':<ref → ''Z''. This is sometimes called the splittingname="SilberschatzKorth2010a"/combining rule.><ref name="Garcia-MolinaUllman2009">{{cite book|author1=Hector Garcia-Molina|author2=Jeffrey D. Ullman|author3=Jennifer Widom|title=Database systems: the complete book|year=2009|publisher=Pearson Prentice Hall|isbn=978-0-13-187325-4|edition=2nd|page=73}} This is sometimes called the splitting/combining rule.</ref>▼
The union and decomposition rules can be combined in a [[logical equivalence]] stating that
▲* '''Decomposition''': If ''X'' → ''YZY'',thenand ''X'' → ''YZ'' [[if and only if]] ''X'' → ''ZYZ''
▲''X'' → ''YZ'', holds [[iff]] ''X'' → ''Y'' and ''X'' → ''Z''. This is sometimes called the splitting/combining rule.<ref name="Garcia-MolinaUllman2009">{{cite book|author1=Hector Garcia-Molina|author2=Jeffrey D. Ullman|author3=Jennifer Widom|title=Database systems: the complete book|year=2009|publisher=Pearson Prentice Hall|isbn=978-0-13-187325-4|edition=2nd|page=73}}</ref>
▲Another rule that is sometimes handy is:<ref name="Singh2009">{{cite book|author=S. K. Singh|title=Database Systems: Concepts, Design & Applications|url=https://books.google.com/books?id=8PNCKe2SpRwC&pg=PA323|year=2009|orig-year=2006|publisher=Pearson Education India|isbn=978-81-7758-567-4|page=323}}</ref>
* '''Composition''': If ''X'' → ''Y'' and ''Z'' → ''W'', then ''XZ'' → ''YW''
