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Theorem difeq1 3204
Description: Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
difeq1  |-  ( A  =  B  ->  ( A  \  C )  =  ( B  \  C
) )

Proof of Theorem difeq1
StepHypRef Expression
1 rabeq 2721 . 2  |-  ( A  =  B  ->  { x  e.  A  |  -.  x  e.  C }  =  { x  e.  B  |  -.  x  e.  C } )
2 dfdif2 3087 . 2  |-  ( A 
\  C )  =  { x  e.  A  |  -.  x  e.  C }
3 dfdif2 3087 . 2  |-  ( B 
\  C )  =  { x  e.  B  |  -.  x  e.  C }
41, 2, 33eqtr4g 2310 1  |-  ( A  =  B  ->  ( A  \  C )  =  ( B  \  C
) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    = wceq 1619    e. wcel 1621   {crab 2512    \ cdif 3075
This theorem is referenced by:  difeq12  3206  difeq1i  3207  difeq1d  3210  uneqdifeq  3448  hartogslem1  7141  kmlem9  7668  kmlem11  7670  kmlem12  7671  isfin1a  7802  fin1a2lem13  7922  ablfac1eulem  15142  islbs  15664  lbsextlem1  15743  lbsextlem2  15744  lbsextlem3  15745  lbsextlem4  15746  lpval  16703  2ndcdisj  17014  isufil  17430  ptcmplem2  17579  mblsplit  18723  voliunlem3  18741  ig1pval  19390  cvmscbv  22960  cvmsi  22967  cvmsval  22968  symdifeq1  23539  sssu  24307  bwt2  24758  f1otrspeq  26556  pmtrval  26560  pmtrfrn  26566  symgsssg  26574  symgfisg  26575  symggen  26577  psgnunilem1  26582  psgnunilem5  26583  psgneldm  26592  sdrgacs  26675  compne  26809
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 1926  ax-ext 2234
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 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-rab 2516  df-dif 3081
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