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Theorem notab 3603
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 2706 . . 3  |-  { x  e.  _V  |  -.  ph }  =  { x  |  ( x  e. 
_V  /\  -.  ph ) }
2 rabab 2965 . . 3  |-  { x  e.  _V  |  -.  ph }  =  { x  |  -.  ph }
31, 2eqtr3i 2457 . 2  |-  { x  |  ( x  e. 
_V  /\  -.  ph ) }  =  { x  |  -.  ph }
4 difab 3602 . . 3  |-  ( { x  |  x  e. 
_V }  \  {
x  |  ph }
)  =  { x  |  ( x  e. 
_V  /\  -.  ph ) }
5 abid2 2552 . . . 4  |-  { x  |  x  e.  _V }  =  _V
65difeq1i 3453 . . 3  |-  ( { x  |  x  e. 
_V }  \  {
x  |  ph }
)  =  ( _V 
\  { x  | 
ph } )
74, 6eqtr3i 2457 . 2  |-  { x  |  ( x  e. 
_V  /\  -.  ph ) }  =  ( _V  \  { x  |  ph } )
83, 7eqtr3i 2457 1  |-  { x  |  -.  ph }  =  ( _V  \  { x  |  ph } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2421   {crab 2701   _Vcvv 2948    \ cdif 3309
This theorem is referenced by:  dfif3  3741
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rab 2706  df-v 2950  df-dif 3315
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