Metamath Proof Explorer

Theorem chsscon3

Description: Hilbert lattice contraposition law. (Contributed by NM, 12-Jun-2004) (New usage is discouraged.)

Ref Expression
Assertion chsscon3 
|- ( ( A e. CH /\ B e. CH ) -> ( A C_ B <-> ( _|_ ` B ) C_ ( _|_ ` A ) ) )

Proof

Step Hyp Ref Expression
1 sseq1 
 |-  ( A = if ( A e. CH , A , ~H ) -> ( A C_ B <-> if ( A e. CH , A , ~H ) C_ B ) )
2 fveq2 
 |-  ( A = if ( A e. CH , A , ~H ) -> ( _|_ ` A ) = ( _|_ ` if ( A e. CH , A , ~H ) ) )
3 2 sseq2d 
 |-  ( A = if ( A e. CH , A , ~H ) -> ( ( _|_ ` B ) C_ ( _|_ ` A ) <-> ( _|_ ` B ) C_ ( _|_ ` if ( A e. CH , A , ~H ) ) ) )
4 1 3 bibi12d 
 |-  ( A = if ( A e. CH , A , ~H ) -> ( ( A C_ B <-> ( _|_ ` B ) C_ ( _|_ ` A ) ) <-> ( if ( A e. CH , A , ~H ) C_ B <-> ( _|_ ` B ) C_ ( _|_ ` if ( A e. CH , A , ~H ) ) ) ) )
5 sseq2 
 |-  ( B = if ( B e. CH , B , ~H ) -> ( if ( A e. CH , A , ~H ) C_ B <-> if ( A e. CH , A , ~H ) C_ if ( B e. CH , B , ~H ) ) )
6 fveq2 
 |-  ( B = if ( B e. CH , B , ~H ) -> ( _|_ ` B ) = ( _|_ ` if ( B e. CH , B , ~H ) ) )
7 6 sseq1d 
 |-  ( B = if ( B e. CH , B , ~H ) -> ( ( _|_ ` B ) C_ ( _|_ ` if ( A e. CH , A , ~H ) ) <-> ( _|_ ` if ( B e. CH , B , ~H ) ) C_ ( _|_ ` if ( A e. CH , A , ~H ) ) ) )
8 5 7 bibi12d 
 |-  ( B = if ( B e. CH , B , ~H ) -> ( ( if ( A e. CH , A , ~H ) C_ B <-> ( _|_ ` B ) C_ ( _|_ ` if ( A e. CH , A , ~H ) ) ) <-> ( if ( A e. CH , A , ~H ) C_ if ( B e. CH , B , ~H ) <-> ( _|_ ` if ( B e. CH , B , ~H ) ) C_ ( _|_ ` if ( A e. CH , A , ~H ) ) ) ) )
9 ifchhv 
 |-  if ( A e. CH , A , ~H ) e. CH
10 ifchhv 
 |-  if ( B e. CH , B , ~H ) e. CH
11 9 10 chsscon3i 
 |-  ( if ( A e. CH , A , ~H ) C_ if ( B e. CH , B , ~H ) <-> ( _|_ ` if ( B e. CH , B , ~H ) ) C_ ( _|_ ` if ( A e. CH , A , ~H ) ) )
12 4 8 11 dedth2h 
 |-  ( ( A e. CH /\ B e. CH ) -> ( A C_ B <-> ( _|_ ` B ) C_ ( _|_ ` A ) ) )