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Theorem notrab 3427
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 3419 . 2  |-  ( { x  |  x  e.  A }  \  {
x  |  ph }
)  =  { x  |  ( x  e.  A  /\  -.  ph ) }
2 difin 3387 . . 3  |-  ( A 
\  ( A  i^i  { x  |  ph }
) )  =  ( A  \  { x  |  ph } )
3 dfrab3 3426 . . . 4  |-  { x  e.  A  |  ph }  =  ( A  i^i  { x  |  ph }
)
43difeq2i 3265 . . 3  |-  ( A 
\  { x  e.  A  |  ph }
)  =  ( A 
\  ( A  i^i  { x  |  ph }
) )
5 abid2 2310 . . . 4  |-  { x  |  x  e.  A }  =  A
65difeq1i 3264 . . 3  |-  ( { x  |  x  e.  A }  \  {
x  |  ph }
)  =  ( A 
\  { x  | 
ph } )
72, 4, 63eqtr4i 2220 . 2  |-  ( A 
\  { x  e.  A  |  ph }
)  =  ( { x  |  x  e.  A }  \  {
x  |  ph }
)
8 df-rab 2477 . 2  |-  { x  e.  A  |  -.  ph }  =  { x  |  ( x  e.  A  /\  -.  ph ) }
91, 7, 83eqtr4i 2220 1  |-  ( A 
\  { x  e.  A  |  ph }
)  =  { x  e.  A  |  -.  ph }
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 104    = wceq 1364    e. wcel 2160   {cab 2175   {crab 2472    \ cdif 3141    i^i cin 3143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rab 2477  df-v 2754  df-dif 3146  df-in 3150
This theorem is referenced by:  diffitest  6916
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