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Theorem notab 3554
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 2658 . . 3  |-  { x  e.  _V  |  -.  ph }  =  { x  |  ( x  e. 
_V  /\  -.  ph ) }
2 rabab 2916 . . 3  |-  { x  e.  _V  |  -.  ph }  =  { x  |  -.  ph }
31, 2eqtr3i 2409 . 2  |-  { x  |  ( x  e. 
_V  /\  -.  ph ) }  =  { x  |  -.  ph }
4 difab 3553 . . 3  |-  ( { x  |  x  e. 
_V }  \  {
x  |  ph }
)  =  { x  |  ( x  e. 
_V  /\  -.  ph ) }
5 abid2 2504 . . . 4  |-  { x  |  x  e.  _V }  =  _V
65difeq1i 3404 . . 3  |-  ( { x  |  x  e. 
_V }  \  {
x  |  ph }
)  =  ( _V 
\  { x  | 
ph } )
74, 6eqtr3i 2409 . 2  |-  { x  |  ( x  e. 
_V  /\  -.  ph ) }  =  ( _V  \  { x  |  ph } )
83, 7eqtr3i 2409 1  |-  { x  |  -.  ph }  =  ( _V  \  { x  |  ph } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 359    = wceq 1649    e. wcel 1717   {cab 2373   {crab 2653   _Vcvv 2899    \ cdif 3260
This theorem is referenced by:  dfif3  3692
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-rab 2658  df-v 2901  df-dif 3266
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