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Theorem compab 27658
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 2578 . . . 4  |-  F/_ z _V
2 nfab1 2580 . . . 4  |-  F/_ z { z  |  ph }
31, 2nfdif 3454 . . 3  |-  F/_ z
( _V  \  {
z  |  ph }
)
4 nfab1 2580 . . 3  |-  F/_ z { z  |  -.  ph }
53, 4cleqf 2602 . 2  |-  ( ( _V  \  { z  |  ph } )  =  { z  |  -.  ph }  <->  A. z
( z  e.  ( _V  \  { z  |  ph } )  <-> 
z  e.  { z  |  -.  ph }
) )
6 abid 2430 . . . 4  |-  ( z  e.  { z  | 
ph }  <->  ph )
76notbii 289 . . 3  |-  ( -.  z  e.  { z  |  ph }  <->  -.  ph )
8 compel 27655 . . 3  |-  ( z  e.  ( _V  \  { z  |  ph } )  <->  -.  z  e.  { z  |  ph } )
9 abid 2430 . . 3  |-  ( z  e.  { z  |  -.  ph }  <->  -.  ph )
107, 8, 93bitr4i 270 . 2  |-  ( z  e.  ( _V  \  { z  |  ph } )  <->  z  e.  { z  |  -.  ph } )
115, 10mpgbir 1560 1  |-  ( _V 
\  { z  | 
ph } )  =  { z  |  -.  ph }
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 178    = wceq 1653    e. wcel 1727   {cab 2428   _Vcvv 2962    \ cdif 3303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-rab 2720  df-v 2964  df-dif 3309
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