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Theorem compab 27611
Description: Two ways of saying "the complement of a class abstraction". (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
Assertion
Ref Expression
compab  |-  ( _V 
\  { z  | 
ph } )  =  { z  |  -.  ph }

Proof of Theorem compab
StepHypRef Expression
1 nfcv 2571 . . . 4  |-  F/_ z _V
2 nfab1 2573 . . . 4  |-  F/_ z { z  |  ph }
31, 2nfdif 3460 . . 3  |-  F/_ z
( _V  \  {
z  |  ph }
)
4 nfab1 2573 . . 3  |-  F/_ z { z  |  -.  ph }
53, 4cleqf 2595 . 2  |-  ( ( _V  \  { z  |  ph } )  =  { z  |  -.  ph }  <->  A. z
( z  e.  ( _V  \  { z  |  ph } )  <-> 
z  e.  { z  |  -.  ph }
) )
6 abid 2423 . . . 4  |-  ( z  e.  { z  | 
ph }  <->  ph )
76notbii 288 . . 3  |-  ( -.  z  e.  { z  |  ph }  <->  -.  ph )
8 compel 27608 . . 3  |-  ( z  e.  ( _V  \  { z  |  ph } )  <->  -.  z  e.  { z  |  ph } )
9 abid 2423 . . 3  |-  ( z  e.  { z  |  -.  ph }  <->  -.  ph )
107, 8, 93bitr4i 269 . 2  |-  ( z  e.  ( _V  \  { z  |  ph } )  <->  z  e.  { z  |  -.  ph } )
115, 10mpgbir 1559 1  |-  ( _V 
\  { z  | 
ph } )  =  { z  |  -.  ph }
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    = wceq 1652    e. wcel 1725   {cab 2421   _Vcvv 2948    \ cdif 3309
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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