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Theorem dmdbr 23755
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 2464 . . . . 5  |-  ( y  =  A  ->  (
y  e.  CH  <->  A  e.  CH ) )
21anbi1d 686 . . . 4  |-  ( y  =  A  ->  (
( y  e.  CH  /\  z  e.  CH )  <->  ( A  e.  CH  /\  z  e.  CH )
) )
3 ineq2 3496 . . . . . . . 8  |-  ( y  =  A  ->  (
x  i^i  y )  =  ( x  i^i 
A ) )
43oveq1d 6055 . . . . . . 7  |-  ( y  =  A  ->  (
( x  i^i  y
)  vH  z )  =  ( ( x  i^i  A )  vH  z ) )
5 oveq1 6047 . . . . . . . 8  |-  ( y  =  A  ->  (
y  vH  z )  =  ( A  vH  z ) )
65ineq2d 3502 . . . . . . 7  |-  ( y  =  A  ->  (
x  i^i  ( y  vH  z ) )  =  ( x  i^i  ( A  vH  z ) ) )
74, 6eqeq12d 2418 . . . . . 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 308 . . . . 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 2686 . . . 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 692 . . 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 2464 . . . . 5  |-  ( z  =  B  ->  (
z  e.  CH  <->  B  e.  CH ) )
1211anbi2d 685 . . . 4  |-  ( z  =  B  ->  (
( A  e.  CH  /\  z  e.  CH )  <->  ( A  e.  CH  /\  B  e.  CH )
) )
13 sseq1 3329 . . . . . 6  |-  ( z  =  B  ->  (
z  C_  x  <->  B  C_  x
) )
14 oveq2 6048 . . . . . . 7  |-  ( z  =  B  ->  (
( x  i^i  A
)  vH  z )  =  ( ( x  i^i  A )  vH  B ) )
15 oveq2 6048 . . . . . . . 8  |-  ( z  =  B  ->  ( A  vH  z )  =  ( A  vH  B
) )
1615ineq2d 3502 . . . . . . 7  |-  ( z  =  B  ->  (
x  i^i  ( A  vH  z ) )  =  ( x  i^i  ( A  vH  B ) ) )
1714, 16eqeq12d 2418 . . . . . 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 312 . . . . 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 2686 . . . 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 692 . . 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 23737 . . 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 4434 . 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 851 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666    i^i cin 3279    C_ wss 3280   class class class wbr 4172  (class class class)co 6040   CHcch 22385    vH chj 22389    MH* cdmd 22423
This theorem is referenced by:  dmdmd  23756  dmdi  23758  dmdbr2  23759  dmdbr3  23761  mddmd2  23765
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 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-iota 5377  df-fv 5421  df-ov 6043  df-dmd 23737
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