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Theorem notrab 3274
Description: Complementation of restricted class abstractions. (Contributed by Mario Carneiro, 3-Sep-2015.)
Assertion
Ref Expression
notrab  |-  ( A 
\  { x  e.  A  |  ph }
)  =  { x  e.  A  |  -.  ph }
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem notrab
StepHypRef Expression
1 difab 3266 . 2  |-  ( { x  |  x  e.  A }  \  {
x  |  ph }
)  =  { x  |  ( x  e.  A  /\  -.  ph ) }
2 difin 3234 . . 3  |-  ( A 
\  ( A  i^i  { x  |  ph }
) )  =  ( A  \  { x  |  ph } )
3 dfrab3 3273 . . . 4  |-  { x  e.  A  |  ph }  =  ( A  i^i  { x  |  ph }
)
43difeq2i 3113 . . 3  |-  ( A 
\  { x  e.  A  |  ph }
)  =  ( A 
\  ( A  i^i  { x  |  ph }
) )
5 abid2 2208 . . . 4  |-  { x  |  x  e.  A }  =  A
65difeq1i 3112 . . 3  |-  ( { x  |  x  e.  A }  \  {
x  |  ph }
)  =  ( A 
\  { x  | 
ph } )
72, 4, 63eqtr4i 2118 . 2  |-  ( A 
\  { x  e.  A  |  ph }
)  =  ( { x  |  x  e.  A }  \  {
x  |  ph }
)
8 df-rab 2368 . 2  |-  { x  e.  A  |  -.  ph }  =  { x  |  ( x  e.  A  /\  -.  ph ) }
91, 7, 83eqtr4i 2118 1  |-  ( A 
\  { x  e.  A  |  ph }
)  =  { x  e.  A  |  -.  ph }
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 102    = wceq 1289    e. wcel 1438   {cab 2074   {crab 2363    \ cdif 2994    i^i cin 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-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-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rab 2368  df-v 2621  df-dif 2999  df-in 3003
This theorem is referenced by:  diffitest  6583
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