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Theorem indif1 3693
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 3692 . 2  |-  ( B  i^i  ( A  \  C ) )  =  ( ( B  i^i  A )  \  C )
2 incom 3631 . 2  |-  ( B  i^i  ( A  \  C ) )  =  ( ( A  \  C )  i^i  B
)
3 incom 3631 . . 3  |-  ( B  i^i  A )  =  ( A  i^i  B
)
43difeq1i 3556 . 2  |-  ( ( B  i^i  A ) 
\  C )  =  ( ( A  i^i  B )  \  C )
51, 2, 43eqtr3i 2439 1  |-  ( ( A  \  C )  i^i  B )  =  ( ( A  i^i  B )  \  C )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1405    \ cdif 3410    i^i cin 3412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2758  df-rab 2762  df-v 3060  df-dif 3416  df-in 3420
This theorem is referenced by:  resdmdfsn  5138  hartogslem1  8000  fpwwe2  9050  leiso  12555  basdif0  19744  tgdif0  19784  kqdisj  20523  trufil  20701  gtiso  27949  dfon4  30218
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