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Theorem dmdbr 23807
Description: Binary relation expressing the dual modular pair property. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
Assertion
Ref Expression
dmdbr  |-  ( ( 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
) ) ) ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem dmdbr
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2498 . . . . 5  |-  ( y  =  A  ->  (
y  e.  CH  <->  A  e.  CH ) )
21anbi1d 687 . . . 4  |-  ( y  =  A  ->  (
( y  e.  CH  /\  z  e.  CH )  <->  ( A  e.  CH  /\  z  e.  CH )
) )
3 ineq2 3538 . . . . . . . 8  |-  ( y  =  A  ->  (
x  i^i  y )  =  ( x  i^i 
A ) )
43oveq1d 6099 . . . . . . 7  |-  ( y  =  A  ->  (
( x  i^i  y
)  vH  z )  =  ( ( x  i^i  A )  vH  z ) )
5 oveq1 6091 . . . . . . . 8  |-  ( y  =  A  ->  (
y  vH  z )  =  ( A  vH  z ) )
65ineq2d 3544 . . . . . . 7  |-  ( y  =  A  ->  (
x  i^i  ( y  vH  z ) )  =  ( x  i^i  ( A  vH  z ) ) )
74, 6eqeq12d 2452 . . . . . 6  |-  ( y  =  A  ->  (
( ( x  i^i  y )  vH  z
)  =  ( x  i^i  ( y  vH  z ) )  <->  ( (
x  i^i  A )  vH  z )  =  ( x  i^i  ( A  vH  z ) ) ) )
87imbi2d 309 . . . . 5  |-  ( y  =  A  ->  (
( z  C_  x  ->  ( ( x  i^i  y )  vH  z
)  =  ( x  i^i  ( y  vH  z ) ) )  <-> 
( z  C_  x  ->  ( ( x  i^i 
A )  vH  z
)  =  ( x  i^i  ( A  vH  z ) ) ) ) )
98ralbidv 2727 . . . 4  |-  ( y  =  A  ->  ( A. x  e.  CH  (
z  C_  x  ->  ( ( x  i^i  y
)  vH  z )  =  ( x  i^i  ( y  vH  z
) ) )  <->  A. x  e.  CH  ( z  C_  x  ->  ( ( x  i^i  A )  vH  z )  =  ( x  i^i  ( A  vH  z ) ) ) ) )
102, 9anbi12d 693 . . 3  |-  ( y  =  A  ->  (
( ( y  e. 
CH  /\  z  e.  CH )  /\  A. x  e.  CH  ( z  C_  x  ->  ( ( x  i^i  y )  vH  z )  =  ( x  i^i  ( y  vH  z ) ) ) )  <->  ( ( A  e.  CH  /\  z  e.  CH )  /\  A. x  e.  CH  ( z 
C_  x  ->  (
( x  i^i  A
)  vH  z )  =  ( x  i^i  ( A  vH  z
) ) ) ) ) )
11 eleq1 2498 . . . . 5  |-  ( z  =  B  ->  (
z  e.  CH  <->  B  e.  CH ) )
1211anbi2d 686 . . . 4  |-  ( z  =  B  ->  (
( A  e.  CH  /\  z  e.  CH )  <->  ( A  e.  CH  /\  B  e.  CH )
) )
13 sseq1 3371 . . . . . 6  |-  ( z  =  B  ->  (
z  C_  x  <->  B  C_  x
) )
14 oveq2 6092 . . . . . . 7  |-  ( z  =  B  ->  (
( x  i^i  A
)  vH  z )  =  ( ( x  i^i  A )  vH  B ) )
15 oveq2 6092 . . . . . . . 8  |-  ( z  =  B  ->  ( A  vH  z )  =  ( A  vH  B
) )
1615ineq2d 3544 . . . . . . 7  |-  ( z  =  B  ->  (
x  i^i  ( A  vH  z ) )  =  ( x  i^i  ( A  vH  B ) ) )
1714, 16eqeq12d 2452 . . . . . 6  |-  ( z  =  B  ->  (
( ( x  i^i 
A )  vH  z
)  =  ( x  i^i  ( A  vH  z ) )  <->  ( (
x  i^i  A )  vH  B )  =  ( x  i^i  ( A  vH  B ) ) ) )
1813, 17imbi12d 313 . . . . 5  |-  ( z  =  B  ->  (
( z  C_  x  ->  ( ( x  i^i 
A )  vH  z
)  =  ( x  i^i  ( A  vH  z ) ) )  <-> 
( B  C_  x  ->  ( ( x  i^i 
A )  vH  B
)  =  ( x  i^i  ( A  vH  B ) ) ) ) )
1918ralbidv 2727 . . . 4  |-  ( z  =  B  ->  ( A. x  e.  CH  (
z  C_  x  ->  ( ( x  i^i  A
)  vH  z )  =  ( x  i^i  ( A  vH  z
) ) )  <->  A. x  e.  CH  ( B  C_  x  ->  ( ( x  i^i  A )  vH  B )  =  ( x  i^i  ( A  vH  B ) ) ) ) )
2012, 19anbi12d 693 . . 3  |-  ( z  =  B  ->  (
( ( A  e. 
CH  /\  z  e.  CH )  /\  A. x  e.  CH  ( z  C_  x  ->  ( ( x  i^i  A )  vH  z )  =  ( x  i^i  ( A  vH  z ) ) ) )  <->  ( ( A  e.  CH  /\  B  e.  CH )  /\  A. x  e.  CH  ( B 
C_  x  ->  (
( x  i^i  A
)  vH  B )  =  ( x  i^i  ( A  vH  B
) ) ) ) ) )
21 df-dmd 23789 . . 3  |-  MH*  =  { <. y ,  z
>.  |  ( (
y  e.  CH  /\  z  e.  CH )  /\  A. x  e.  CH  ( z  C_  x  ->  ( ( x  i^i  y )  vH  z
)  =  ( x  i^i  ( y  vH  z ) ) ) ) }
2210, 20, 21brabg 4477 . 2  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH*  B  <->  ( ( A  e.  CH  /\  B  e.  CH )  /\  A. x  e.  CH  ( B  C_  x  -> 
( ( x  i^i 
A )  vH  B
)  =  ( x  i^i  ( A  vH  B ) ) ) ) ) )
2322bianabs 852 1  |-  ( ( 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
) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = 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:  dmdmd  23808  dmdi  23810  dmdbr2  23811  dmdbr3  23813  mddmd2  23817
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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