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Theorem indif1 3426
Description: Bring an intersection in and out of a class difference. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
indif1  |-  ( ( A  \  C )  i^i  B )  =  ( ( A  i^i  B )  \  C )

Proof of Theorem indif1
StepHypRef Expression
1 indif2 3425 . 2  |-  ( B  i^i  ( A  \  C ) )  =  ( ( B  i^i  A )  \  C )
2 incom 3374 . 2  |-  ( B  i^i  ( A  \  C ) )  =  ( ( A  \  C )  i^i  B
)
3 incom 3374 . . 3  |-  ( B  i^i  A )  =  ( A  i^i  B
)
43difeq1i 3303 . 2  |-  ( ( B  i^i  A ) 
\  C )  =  ( ( A  i^i  B )  \  C )
51, 2, 43eqtr3i 2324 1  |-  ( ( A  \  C )  i^i  B )  =  ( ( A  i^i  B )  \  C )
Colors of variables: wff set class
Syntax hints:    = wceq 1632    \ cdif 3162    i^i cin 3164
This theorem is referenced by:  hartogslem1  7273  fpwwe2  8281  leiso  11413  basdif0  16707  tgdif0  16746  kqdisj  17439  trufil  17621  gtiso  23256  dfon4  24504
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
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