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Theorem dmdi 23762
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 23759 . . . . 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 199 . . . 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 3334 . . . . . 6  |-  ( x  =  C  ->  ( B  C_  x  <->  B  C_  C
) )
4 ineq1 3499 . . . . . . . 8  |-  ( x  =  C  ->  (
x  i^i  A )  =  ( C  i^i  A ) )
54oveq1d 6059 . . . . . . 7  |-  ( x  =  C  ->  (
( x  i^i  A
)  vH  B )  =  ( ( C  i^i  A )  vH  B ) )
6 ineq1 3499 . . . . . . 7  |-  ( x  =  C  ->  (
x  i^i  ( A  vH  B ) )  =  ( C  i^i  ( A  vH  B ) ) )
75, 6eqeq12d 2422 . . . . . 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 312 . . . . 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 3012 . . . 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 639 . . 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 1148 . 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 423 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2670    i^i cin 3283    C_ wss 3284   class class class wbr 4176  (class class class)co 6044   CHcch 22389    vH chj 22393    MH* cdmd 22427
This theorem is referenced by:  dmdi2  23764  dmdsl3  23775  csmdsymi  23794  mdsymlem1  23863
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-iota 5381  df-fv 5425  df-ov 6047  df-dmd 23741
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