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Theorem difeq2 3112
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 2151 . . . 4  |-  ( A  =  B  ->  (
x  e.  A  <->  x  e.  B ) )
21notbid 627 . . 3  |-  ( A  =  B  ->  ( -.  x  e.  A  <->  -.  x  e.  B ) )
32rabbidv 2608 . 2  |-  ( A  =  B  ->  { x  e.  C  |  -.  x  e.  A }  =  { x  e.  C  |  -.  x  e.  B } )
4 dfdif2 3007 . 2  |-  ( C 
\  A )  =  { x  e.  C  |  -.  x  e.  A }
5 dfdif2 3007 . 2  |-  ( C 
\  B )  =  { x  e.  C  |  -.  x  e.  B }
63, 4, 53eqtr4g 2145 1  |-  ( A  =  B  ->  ( C  \  A )  =  ( C  \  B
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1289    e. wcel 1438   {crab 2363    \ cdif 2996
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-in1 579  ax-in2 580  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  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-ral 2364  df-rab 2368  df-dif 3001
This theorem is referenced by:  difeq12  3113  difeq2i  3115  difeq2d  3118  disjdif2  3361  ssdifeq0  3365  2oconcl  6203  diffitest  6603  diffifi  6610  undifdc  6634  sbthlem2  6667  isbth  6676
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