Metamath Proof Explorer

Theorem chsscon2

Description: Hilbert lattice contraposition law. (Contributed by NM, 21-Jun-2004) (New usage is discouraged.)

Ref Expression
Assertion chsscon2 
|- ( ( A e. CH /\ B e. CH ) -> ( A C_ ( _|_ ` B ) <-> B C_ ( _|_ ` A ) ) )

Proof

Step Hyp Ref Expression
1 chss 
 |-  ( A e. CH -> A C_ ~H )
2 chss 
 |-  ( B e. CH -> B C_ ~H )
3 occon3 
 |-  ( ( A C_ ~H /\ B C_ ~H ) -> ( A C_ ( _|_ ` B ) <-> B C_ ( _|_ ` A ) ) )
4 1 2 3 syl2an 
 |-  ( ( A e. CH /\ B e. CH ) -> ( A C_ ( _|_ ` B ) <-> B C_ ( _|_ ` A ) ) )