Metamath Proof Explorer

Theorem mdsym

Description: M-symmetry of the Hilbert lattice. Lemma 5 of Maeda p. 168. (Contributed by NM, 6-Jul-2004) (New usage is discouraged.)

Ref Expression
Assertion mdsym 
|- ( ( A e. CH /\ B e. CH ) -> ( A MH B <-> B MH A ) )

Proof

Step Hyp Ref Expression
1 breq1 
 |-  ( A = if ( A e. CH , A , ~H ) -> ( A MH B <-> if ( A e. CH , A , ~H ) MH B ) )
2 breq2 
 |-  ( A = if ( A e. CH , A , ~H ) -> ( B MH A <-> B MH if ( A e. CH , A , ~H ) ) )
3 1 2 bibi12d 
 |-  ( A = if ( A e. CH , A , ~H ) -> ( ( A MH B <-> B MH A ) <-> ( if ( A e. CH , A , ~H ) MH B <-> B MH if ( A e. CH , A , ~H ) ) ) )
4 breq2 
 |-  ( B = if ( B e. CH , B , ~H ) -> ( if ( A e. CH , A , ~H ) MH B <-> if ( A e. CH , A , ~H ) MH if ( B e. CH , B , ~H ) ) )
5 breq1 
 |-  ( B = if ( B e. CH , B , ~H ) -> ( B MH if ( A e. CH , A , ~H ) <-> if ( B e. CH , B , ~H ) MH if ( A e. CH , A , ~H ) ) )
6 4 5 bibi12d 
 |-  ( B = if ( B e. CH , B , ~H ) -> ( ( if ( A e. CH , A , ~H ) MH B <-> B MH if ( A e. CH , A , ~H ) ) <-> ( if ( A e. CH , A , ~H ) MH if ( B e. CH , B , ~H ) <-> if ( B e. CH , B , ~H ) MH if ( A e. CH , A , ~H ) ) ) )
7 ifchhv 
 |-  if ( A e. CH , A , ~H ) e. CH
8 ifchhv 
 |-  if ( B e. CH , B , ~H ) e. CH
9 7 8 mdsymi 
 |-  ( if ( A e. CH , A , ~H ) MH if ( B e. CH , B , ~H ) <-> if ( B e. CH , B , ~H ) MH if ( A e. CH , A , ~H ) )
10 3 6 9 dedth2h 
 |-  ( ( A e. CH /\ B e. CH ) -> ( A MH B <-> B MH A ) )