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Theorem notab 3250
Description: A class builder defined by a negation. (Contributed by FL, 18-Sep-2010.)
Assertion
Ref Expression
notab  |-  { x  |  -.  ph }  =  ( _V  \  { x  |  ph } )

Proof of Theorem notab
StepHypRef Expression
1 df-rab 2362 . . 3  |-  { x  e.  _V  |  -.  ph }  =  { x  |  ( x  e. 
_V  /\  -.  ph ) }
2 rabab 2629 . . 3  |-  { x  e.  _V  |  -.  ph }  =  { x  |  -.  ph }
31, 2eqtr3i 2105 . 2  |-  { x  |  ( x  e. 
_V  /\  -.  ph ) }  =  { x  |  -.  ph }
4 difab 3249 . . 3  |-  ( { x  |  x  e. 
_V }  \  {
x  |  ph }
)  =  { x  |  ( x  e. 
_V  /\  -.  ph ) }
5 abid2 2203 . . . 4  |-  { x  |  x  e.  _V }  =  _V
65difeq1i 3096 . . 3  |-  ( { x  |  x  e. 
_V }  \  {
x  |  ph }
)  =  ( _V 
\  { x  | 
ph } )
74, 6eqtr3i 2105 . 2  |-  { x  |  ( x  e. 
_V  /\  -.  ph ) }  =  ( _V  \  { x  |  ph } )
83, 7eqtr3i 2105 1  |-  { x  |  -.  ph }  =  ( _V  \  { x  |  ph } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 102    = wceq 1285    e. wcel 1434   {cab 2069   {crab 2357   _Vcvv 2610    \ cdif 2979
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rab 2362  df-v 2612  df-dif 2984
This theorem is referenced by:  dfif3  3381
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