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Theorem indif 3353
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 3351 . 2  |-  ( A  i^i  ( A  \  B ) )  =  ( A  \  ( A  \  ( A  \  B ) ) )
2 dfin4 3351 . . 3  |-  ( A  i^i  B )  =  ( A  \  ( A  \  B ) )
32difeq2i 3233 . 2  |-  ( A 
\  ( A  i^i  B ) )  =  ( A  \  ( A 
\  ( A  \  B ) ) )
4 difin 3348 . 2  |-  ( A 
\  ( A  i^i  B ) )  =  ( A  \  B )
51, 3, 43eqtr2i 2282 1  |-  ( A  i^i  ( A  \  B ) )  =  ( A  \  B
)
Colors of variables: wff set class
Syntax hints:    = wceq 1619    \ cdif 3091    i^i cin 3093
This theorem is referenced by:  resdif  5397  kmlem11  7719
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ral 2520  df-rab 2523  df-v 2742  df-dif 3097  df-in 3101  df-ss 3108
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