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Theorem indif 3412
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 3410 . 2  |-  ( A  i^i  ( A  \  B ) )  =  ( A  \  ( A  \  ( A  \  B ) ) )
2 dfin4 3410 . . 3  |-  ( A  i^i  B )  =  ( A  \  ( A  \  B ) )
32difeq2i 3292 . 2  |-  ( A 
\  ( A  i^i  B ) )  =  ( A  \  ( A 
\  ( A  \  B ) ) )
4 difin 3407 . 2  |-  ( A 
\  ( A  i^i  B ) )  =  ( A  \  B )
51, 3, 43eqtr2i 2310 1  |-  ( A  i^i  ( A  \  B ) )  =  ( A  \  B
)
Colors of variables: wff set class
Syntax hints:    = wceq 1628    \ cdif 3150    i^i cin 3152
This theorem is referenced by:  resdif  5459  kmlem11  7781
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ral 2549  df-rab 2553  df-v 2791  df-dif 3156  df-in 3160  df-ss 3167
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