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Theorem compab 27644
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 2419 . . . 4  |-  F/_ z _V
2 nfab1 2421 . . . 4  |-  F/_ z { z  |  ph }
31, 2nfdif 3297 . . 3  |-  F/_ z
( _V  \  {
z  |  ph }
)
4 nfab1 2421 . . 3  |-  F/_ z { z  |  -.  ph }
53, 4cleqf 2443 . 2  |-  ( ( _V  \  { z  |  ph } )  =  { z  |  -.  ph }  <->  A. z
( z  e.  ( _V  \  { z  |  ph } )  <-> 
z  e.  { z  |  -.  ph }
) )
6 abid 2271 . . . 4  |-  ( z  e.  { z  | 
ph }  <->  ph )
76notbii 287 . . 3  |-  ( -.  z  e.  { z  |  ph }  <->  -.  ph )
8 compel 27640 . . 3  |-  ( z  e.  ( _V  \  { z  |  ph } )  <->  -.  z  e.  { z  |  ph } )
9 abid 2271 . . 3  |-  ( z  e.  { z  |  -.  ph }  <->  -.  ph )
107, 8, 93bitr4i 268 . 2  |-  ( z  e.  ( _V  \  { z  |  ph } )  <->  z  e.  { z  |  -.  ph } )
115, 10mpgbir 1537 1  |-  ( _V 
\  { z  | 
ph } )  =  { z  |  -.  ph }
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    = wceq 1623    e. wcel 1684   {cab 2269   _Vcvv 2788    \ cdif 3149
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-dif 3155
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