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Theorem dmdi 22990
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 22987 . . . . 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 3276 . . . . . 6  |-  ( x  =  C  ->  ( B  C_  x  <->  B  C_  C
) )
4 ineq1 3439 . . . . . . . 8  |-  ( x  =  C  ->  (
x  i^i  A )  =  ( C  i^i  A ) )
54oveq1d 5957 . . . . . . 7  |-  ( x  =  C  ->  (
( x  i^i  A
)  vH  B )  =  ( ( C  i^i  A )  vH  B ) )
6 ineq1 3439 . . . . . . 7  |-  ( x  =  C  ->  (
x  i^i  ( A  vH  B ) )  =  ( C  i^i  ( A  vH  B ) ) )
75, 6eqeq12d 2372 . . . . . 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 2956 . . . 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 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:  dmdi2  22992  dmdsl3  23003  csmdsymi  23022  mdsymlem1  23091
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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