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Theorem mdi 25634
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 25633 . . . . 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 207 . . . 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 3374 . . . . . 6  |-  ( x  =  C  ->  (
x  C_  B  <->  C  C_  B
) )
4 oveq1 6097 . . . . . . . 8  |-  ( x  =  C  ->  (
x  vH  A )  =  ( C  vH  A ) )
54ineq1d 3548 . . . . . . 7  |-  ( x  =  C  ->  (
( x  vH  A
)  i^i  B )  =  ( ( C  vH  A )  i^i 
B ) )
6 oveq1 6097 . . . . . . 7  |-  ( x  =  C  ->  (
x  vH  ( A  i^i  B ) )  =  ( C  vH  ( A  i^i  B ) ) )
75, 6eqeq12d 2455 . . . . . 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 320 . . . . 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 3066 . . . 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 652 . . 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 1177 . 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 433 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   A.wral 2713    i^i cin 3324    C_ wss 3325   class class class wbr 4289  (class class class)co 6090   CHcch 24266    vH chj 24270    MH cmd 24303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-iota 5378  df-fv 5423  df-ov 6093  df-md 25619
This theorem is referenced by:  mdsl3  25655  mdslmd3i  25671  mdexchi  25674  atabsi  25740
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