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Theorem dmdi 23810
Description: Consequence of the dual modular pair property. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
Assertion
Ref Expression
dmdi  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  MH*  B  /\  B  C_  C ) )  ->  ( ( C  i^i  A )  vH  B )  =  ( C  i^i  ( A  vH  B ) ) )

Proof of Theorem dmdi
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dmdbr 23807 . . . . 5  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH*  B  <->  A. x  e.  CH  ( B  C_  x  ->  (
( x  i^i  A
)  vH  B )  =  ( x  i^i  ( A  vH  B
) ) ) ) )
21biimpd 200 . . . 4  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH*  B  ->  A. x  e.  CH  ( B  C_  x  -> 
( ( x  i^i 
A )  vH  B
)  =  ( x  i^i  ( A  vH  B ) ) ) ) )
3 sseq2 3372 . . . . . 6  |-  ( x  =  C  ->  ( B  C_  x  <->  B  C_  C
) )
4 ineq1 3537 . . . . . . . 8  |-  ( x  =  C  ->  (
x  i^i  A )  =  ( C  i^i  A ) )
54oveq1d 6099 . . . . . . 7  |-  ( x  =  C  ->  (
( x  i^i  A
)  vH  B )  =  ( ( C  i^i  A )  vH  B ) )
6 ineq1 3537 . . . . . . 7  |-  ( x  =  C  ->  (
x  i^i  ( A  vH  B ) )  =  ( C  i^i  ( A  vH  B ) ) )
75, 6eqeq12d 2452 . . . . . 6  |-  ( x  =  C  ->  (
( ( x  i^i 
A )  vH  B
)  =  ( x  i^i  ( A  vH  B ) )  <->  ( ( C  i^i  A )  vH  B )  =  ( C  i^i  ( A  vH  B ) ) ) )
83, 7imbi12d 313 . . . . 5  |-  ( x  =  C  ->  (
( B  C_  x  ->  ( ( x  i^i 
A )  vH  B
)  =  ( x  i^i  ( A  vH  B ) ) )  <-> 
( B  C_  C  ->  ( ( C  i^i  A )  vH  B )  =  ( C  i^i  ( A  vH  B ) ) ) ) )
98rspcv 3050 . . . 4  |-  ( C  e.  CH  ->  ( A. x  e.  CH  ( B  C_  x  ->  (
( x  i^i  A
)  vH  B )  =  ( x  i^i  ( A  vH  B
) ) )  -> 
( B  C_  C  ->  ( ( C  i^i  A )  vH  B )  =  ( C  i^i  ( A  vH  B ) ) ) ) )
102, 9sylan9 640 . . 3  |-  ( ( ( A  e.  CH  /\  B  e.  CH )  /\  C  e.  CH )  ->  ( A  MH*  B  ->  ( B  C_  C  ->  ( ( C  i^i  A )  vH  B )  =  ( C  i^i  ( A  vH  B ) ) ) ) )
11103impa 1149 . 2  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  MH*  B  ->  ( B  C_  C  ->  (
( C  i^i  A
)  vH  B )  =  ( C  i^i  ( A  vH  B ) ) ) ) )
1211imp32 424 1  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  MH*  B  /\  B  C_  C ) )  ->  ( ( C  i^i  A )  vH  B )  =  ( C  i^i  ( A  vH  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707    i^i cin 3321    C_ wss 3322   class class class wbr 4215  (class class class)co 6084   CHcch 22437    vH chj 22441    MH* cdmd 22475
This theorem is referenced by:  dmdi2  23812  dmdsl3  23823  csmdsymi  23842  mdsymlem1  23911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-iota 5421  df-fv 5465  df-ov 6087  df-dmd 23789
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