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Theorem mdi 22871
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 22870 . . . . 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 3200 . . . . . 6  |-  ( x  =  C  ->  (
x  C_  B  <->  C  C_  B
) )
4 oveq1 5827 . . . . . . . 8  |-  ( x  =  C  ->  (
x  vH  A )  =  ( C  vH  A ) )
54ineq1d 3370 . . . . . . 7  |-  ( x  =  C  ->  (
( x  vH  A
)  i^i  B )  =  ( ( C  vH  A )  i^i 
B ) )
6 oveq1 5827 . . . . . . 7  |-  ( x  =  C  ->  (
x  vH  ( A  i^i  B ) )  =  ( C  vH  ( A  i^i  B ) ) )
75, 6eqeq12d 2298 . . . . . 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 2881 . . . 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 1623    e. wcel 1685   A.wral 2544    i^i cin 3152    C_ wss 3153   class class class wbr 4024  (class class class)co 5820   CHcch 21505    vH chj 21509    MH cmd 21542
This theorem is referenced by:  mdsl3  22892  mdslmd3i  22908  mdexchi  22911  atabsi  22977
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-xp 4694  df-cnv 4696  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fv 5229  df-ov 5823  df-md 22856
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