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Theorem indif 3575
Description: Intersection with class difference. Theorem 34 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
indif  |-  ( A  i^i  ( A  \  B ) )  =  ( A  \  B
)

Proof of Theorem indif
StepHypRef Expression
1 dfin4 3573 . 2  |-  ( A  i^i  ( A  \  B ) )  =  ( A  \  ( A  \  ( A  \  B ) ) )
2 dfin4 3573 . . 3  |-  ( A  i^i  B )  =  ( A  \  ( A  \  B ) )
32difeq2i 3454 . 2  |-  ( A 
\  ( A  i^i  B ) )  =  ( A  \  ( A 
\  ( A  \  B ) ) )
4 difin 3570 . 2  |-  ( A 
\  ( A  i^i  B ) )  =  ( A  \  B )
51, 3, 43eqtr2i 2461 1  |-  ( A  i^i  ( A  \  B ) )  =  ( A  \  B
)
Colors of variables: wff set class
Syntax hints:    = wceq 1652    \ cdif 3309    i^i cin 3311
This theorem is referenced by:  resdif  5687  kmlem11  8029
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rab 2706  df-v 2950  df-dif 3315  df-in 3319  df-ss 3326
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