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Theorem dmdi2 23807
Description: Consequence of the dual modular pair property. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)
Assertion
Ref Expression
dmdi2  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  MH*  B  /\  B  C_  C ) )  ->  ( C  i^i  ( A  vH  B
) )  C_  (
( C  i^i  A
)  vH  B )
)

Proof of Theorem dmdi2
StepHypRef Expression
1 dmdi 23805 . 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 ) ) )
2 eqimss2 3401 . 2  |-  ( ( ( C  i^i  A
)  vH  B )  =  ( C  i^i  ( A  vH  B ) )  ->  ( C  i^i  ( A  vH  B
) )  C_  (
( C  i^i  A
)  vH  B )
)
31, 2syl 16 1  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  MH*  B  /\  B  C_  C ) )  ->  ( C  i^i  ( A  vH  B
) )  C_  (
( C  i^i  A
)  vH  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    i^i cin 3319    C_ wss 3320   class class class wbr 4212  (class class class)co 6081   CHcch 22432    vH chj 22436    MH* cdmd 22470
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-iota 5418  df-fv 5462  df-ov 6084  df-dmd 23784
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