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Theorem dmdbr 22987
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 2418 . . . . 5  |-  ( y  =  A  ->  (
y  e.  CH  <->  A  e.  CH ) )
21anbi1d 685 . . . 4  |-  ( y  =  A  ->  (
( y  e.  CH  /\  z  e.  CH )  <->  ( A  e.  CH  /\  z  e.  CH )
) )
3 ineq2 3440 . . . . . . . 8  |-  ( y  =  A  ->  (
x  i^i  y )  =  ( x  i^i 
A ) )
43oveq1d 5957 . . . . . . 7  |-  ( y  =  A  ->  (
( x  i^i  y
)  vH  z )  =  ( ( x  i^i  A )  vH  z ) )
5 oveq1 5949 . . . . . . . 8  |-  ( y  =  A  ->  (
y  vH  z )  =  ( A  vH  z ) )
65ineq2d 3446 . . . . . . 7  |-  ( y  =  A  ->  (
x  i^i  ( y  vH  z ) )  =  ( x  i^i  ( A  vH  z ) ) )
74, 6eqeq12d 2372 . . . . . 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 307 . . . . 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 2639 . . . 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 691 . . 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 2418 . . . . 5  |-  ( z  =  B  ->  (
z  e.  CH  <->  B  e.  CH ) )
1211anbi2d 684 . . . 4  |-  ( z  =  B  ->  (
( A  e.  CH  /\  z  e.  CH )  <->  ( A  e.  CH  /\  B  e.  CH )
) )
13 sseq1 3275 . . . . . 6  |-  ( z  =  B  ->  (
z  C_  x  <->  B  C_  x
) )
14 oveq2 5950 . . . . . . 7  |-  ( z  =  B  ->  (
( x  i^i  A
)  vH  z )  =  ( ( x  i^i  A )  vH  B ) )
15 oveq2 5950 . . . . . . . 8  |-  ( z  =  B  ->  ( A  vH  z )  =  ( A  vH  B
) )
1615ineq2d 3446 . . . . . . 7  |-  ( z  =  B  ->  (
x  i^i  ( A  vH  z ) )  =  ( x  i^i  ( A  vH  B ) ) )
1714, 16eqeq12d 2372 . . . . . 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 311 . . . . 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 2639 . . . 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 691 . . 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 22969 . . 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 4363 . 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 850 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 176    /\ wa 358    = wceq 1642    e. wcel 1710   A.wral 2619    i^i cin 3227    C_ wss 3228   class class class wbr 4102  (class class class)co 5942   CHcch 21617    vH chj 21621    MH* cdmd 21655
This theorem is referenced by:  dmdmd  22988  dmdi  22990  dmdbr2  22991  dmdbr3  22993  mddmd2  22997
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-iota 5298  df-fv 5342  df-ov 5945  df-dmd 22969
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