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Theorem difeq1 3109
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
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 rabeq 2611 . 2  |-  ( A  =  B  ->  { x  e.  A  |  -.  x  e.  C }  =  { x  e.  B  |  -.  x  e.  C } )
2 dfdif2 3005 . 2  |-  ( A 
\  C )  =  { x  e.  A  |  -.  x  e.  C }
3 dfdif2 3005 . 2  |-  ( B 
\  C )  =  { x  e.  B  |  -.  x  e.  C }
41, 2, 33eqtr4g 2145 1  |-  ( A  =  B  ->  ( A  \  C )  =  ( B  \  C
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1289    e. wcel 1438   {crab 2363    \ cdif 2994
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rab 2368  df-dif 2999
This theorem is referenced by:  difeq12  3111  difeq1i  3112  difeq1d  3115  uneqdifeqim  3364  diffitest  6583
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