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Theorem dmdi 22882
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 22879 . . . . 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 198 . . . 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 3200 . . . . . 6  |-  ( x  =  C  ->  ( B  C_  x  <->  B  C_  C
) )
4 ineq1 3363 . . . . . . . 8  |-  ( x  =  C  ->  (
x  i^i  A )  =  ( C  i^i  A ) )
54oveq1d 5873 . . . . . . 7  |-  ( x  =  C  ->  (
( x  i^i  A
)  vH  B )  =  ( ( C  i^i  A )  vH  B ) )
6 ineq1 3363 . . . . . . 7  |-  ( x  =  C  ->  (
x  i^i  ( A  vH  B ) )  =  ( C  i^i  ( A  vH  B ) ) )
75, 6eqeq12d 2297 . . . . . 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 311 . . . . 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 2880 . . . 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 638 . . 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 1146 . 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 422 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    i^i cin 3151    C_ wss 3152   class class class wbr 4023  (class class class)co 5858   CHcch 21509    vH chj 21513    MH* cdmd 21547
This theorem is referenced by:  dmdi2  22884  dmdsl3  22895  csmdsymi  22914  mdsymlem1  22983
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-iota 5219  df-fv 5263  df-ov 5861  df-dmd 22861
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