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Theorem indif 3424
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 3422 . 2  |-  ( A  i^i  ( A  \  B ) )  =  ( A  \  ( A  \  ( A  \  B ) ) )
2 dfin4 3422 . . 3  |-  ( A  i^i  B )  =  ( A  \  ( A  \  B ) )
32difeq2i 3304 . 2  |-  ( A 
\  ( A  i^i  B ) )  =  ( A  \  ( A 
\  ( A  \  B ) ) )
4 difin 3419 . 2  |-  ( A 
\  ( A  i^i  B ) )  =  ( A  \  B )
51, 3, 43eqtr2i 2322 1  |-  ( A  i^i  ( A  \  B ) )  =  ( A  \  B
)
Colors of variables: wff set class
Syntax hints:    = wceq 1632    \ cdif 3162    i^i cin 3164
This theorem is referenced by:  resdif  5510  kmlem11  7802
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rab 2565  df-v 2803  df-dif 3168  df-in 3172  df-ss 3179
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