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Theorem mdi 22877
Description: Consequence of the modular pair property. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
mdi  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  MH  B  /\  C  C_  B ) )  ->  ( ( C  vH  A )  i^i 
B )  =  ( C  vH  ( A  i^i  B ) ) )

Proof of Theorem mdi
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 mdbr 22876 . . . . 5  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH  B  <->  A. x  e.  CH  (
x  C_  B  ->  ( ( x  vH  A
)  i^i  B )  =  ( x  vH  ( A  i^i  B ) ) ) ) )
21biimpd 198 . . . 4  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH  B  ->  A. x  e.  CH  ( x  C_  B  -> 
( ( x  vH  A )  i^i  B
)  =  ( x  vH  ( A  i^i  B ) ) ) ) )
3 sseq1 3201 . . . . . 6  |-  ( x  =  C  ->  (
x  C_  B  <->  C  C_  B
) )
4 oveq1 5867 . . . . . . . 8  |-  ( x  =  C  ->  (
x  vH  A )  =  ( C  vH  A ) )
54ineq1d 3371 . . . . . . 7  |-  ( x  =  C  ->  (
( x  vH  A
)  i^i  B )  =  ( ( C  vH  A )  i^i 
B ) )
6 oveq1 5867 . . . . . . 7  |-  ( x  =  C  ->  (
x  vH  ( A  i^i  B ) )  =  ( C  vH  ( A  i^i  B ) ) )
75, 6eqeq12d 2299 . . . . . 6  |-  ( x  =  C  ->  (
( ( x  vH  A )  i^i  B
)  =  ( x  vH  ( A  i^i  B ) )  <->  ( ( C  vH  A )  i^i 
B )  =  ( C  vH  ( A  i^i  B ) ) ) )
83, 7imbi12d 311 . . . . 5  |-  ( x  =  C  ->  (
( x  C_  B  ->  ( ( x  vH  A )  i^i  B
)  =  ( x  vH  ( A  i^i  B ) ) )  <->  ( C  C_  B  ->  ( ( C  vH  A )  i^i 
B )  =  ( C  vH  ( A  i^i  B ) ) ) ) )
98rspcv 2882 . . . 4  |-  ( C  e.  CH  ->  ( A. x  e.  CH  (
x  C_  B  ->  ( ( x  vH  A
)  i^i  B )  =  ( x  vH  ( A  i^i  B ) ) )  ->  ( C  C_  B  ->  (
( C  vH  A
)  i^i  B )  =  ( C  vH  ( A  i^i  B ) ) ) ) )
102, 9sylan9 638 . . 3  |-  ( ( ( A  e.  CH  /\  B  e.  CH )  /\  C  e.  CH )  ->  ( A  MH  B  ->  ( C  C_  B  ->  ( ( C  vH  A )  i^i  B
)  =  ( C  vH  ( A  i^i  B ) ) ) ) )
11103impa 1146 . 2  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  MH  B  ->  ( C  C_  B  ->  ( ( C  vH  A
)  i^i  B )  =  ( C  vH  ( A  i^i  B ) ) ) ) )
1211imp32 422 1  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  MH  B  /\  C  C_  B ) )  ->  ( ( C  vH  A )  i^i 
B )  =  ( C  vH  ( A  i^i  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686   A.wral 2545    i^i cin 3153    C_ wss 3154   class class class wbr 4025  (class class class)co 5860   CHcch 21511    vH chj 21515    MH cmd 21548
This theorem is referenced by:  mdsl3  22898  mdslmd3i  22914  mdexchi  22917  atabsi  22983
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-iota 5221  df-fv 5265  df-ov 5863  df-md 22862
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