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Theorem difeq2 3158
Description: Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
difeq2  |-  ( A  =  B  ->  ( C  \  A )  =  ( C  \  B
) )

Proof of Theorem difeq2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eleq2 2181 . . . 4  |-  ( A  =  B  ->  (
x  e.  A  <->  x  e.  B ) )
21notbid 641 . . 3  |-  ( A  =  B  ->  ( -.  x  e.  A  <->  -.  x  e.  B ) )
32rabbidv 2649 . 2  |-  ( A  =  B  ->  { x  e.  C  |  -.  x  e.  A }  =  { x  e.  C  |  -.  x  e.  B } )
4 dfdif2 3049 . 2  |-  ( C 
\  A )  =  { x  e.  C  |  -.  x  e.  A }
5 dfdif2 3049 . 2  |-  ( C 
\  B )  =  { x  e.  C  |  -.  x  e.  B }
63, 4, 53eqtr4g 2175 1  |-  ( A  =  B  ->  ( C  \  A )  =  ( C  \  B
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1316    e. wcel 1465   {crab 2397    \ cdif 3038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-ral 2398  df-rab 2402  df-dif 3043
This theorem is referenced by:  difeq12  3159  difeq2i  3161  difeq2d  3164  disjdif2  3411  ssdifeq0  3415  2oconcl  6304  diffitest  6749  diffifi  6756  undifdc  6780  sbthlem2  6814  isbth  6823  difinfinf  6954  ismkvnex  6997  iscld  12199
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