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Metamath Proof Explorer

Theorem cvmd

Description: The covering property implies the modular pair property. Lemma 7.5.1 of MaedaMaeda p. 31. (Contributed by NM, 21-Jun-2004) (New usage is discouraged.)

Ref Expression
Assertion cvmd 
|- ( ( A e. CH /\ B e. CH /\ ( A i^i B )  A MH B )

Proof

Step Hyp Ref Expression
1 ineq1 
 |-  ( A = if ( A e. CH , A , ~H ) -> ( A i^i B ) = ( if ( A e. CH , A , ~H ) i^i B ) )
2 1 breq1d 
 |-  ( A = if ( A e. CH , A , ~H ) -> ( ( A i^i B )  ( if ( A e. CH , A , ~H ) i^i B )
3 breq1 
 |-  ( A = if ( A e. CH , A , ~H ) -> ( A MH B <-> if ( A e. CH , A , ~H ) MH B ) )
4 2 3 imbi12d 
 |-  ( A = if ( A e. CH , A , ~H ) -> ( ( ( A i^i B )  A MH B ) <-> ( ( if ( A e. CH , A , ~H ) i^i B )  if ( A e. CH , A , ~H ) MH B ) ) )
5 ineq2 
 |-  ( B = if ( B e. CH , B , ~H ) -> ( if ( A e. CH , A , ~H ) i^i B ) = ( if ( A e. CH , A , ~H ) i^i if ( B e. CH , B , ~H ) ) )
6 id 
 |-  ( B = if ( B e. CH , B , ~H ) -> B = if ( B e. CH , B , ~H ) )
7 5 6 breq12d 
 |-  ( B = if ( B e. CH , B , ~H ) -> ( ( if ( A e. CH , A , ~H ) i^i B )  ( if ( A e. CH , A , ~H ) i^i if ( B e. CH , B , ~H ) )
8 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 ) ) )
9 7 8 imbi12d 
 |-  ( B = if ( B e. CH , B , ~H ) -> ( ( ( if ( A e. CH , A , ~H ) i^i B )  if ( A e. CH , A , ~H ) MH B ) <-> ( ( if ( A e. CH , A , ~H ) i^i if ( B e. CH , B , ~H ) )  if ( A e. CH , A , ~H ) MH if ( B e. CH , B , ~H ) ) ) )
10 ifchhv 
 |-  if ( A e. CH , A , ~H ) e. CH
11 ifchhv 
 |-  if ( B e. CH , B , ~H ) e. CH
12 10 11 cvmdi 
 |-  ( ( if ( A e. CH , A , ~H ) i^i if ( B e. CH , B , ~H ) )  if ( A e. CH , A , ~H ) MH if ( B e. CH , B , ~H ) )
13 4 9 12 dedth2h 
 |-  ( ( A e. CH /\ B e. CH ) -> ( ( A i^i B )  A MH B ) )
14 13 3impia 
 |-  ( ( A e. CH /\ B e. CH /\ ( A i^i B )  A MH B )