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Theorem compab 27747
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 2432 . . . 4  |-  F/_ z _V
2 nfab1 2434 . . . 4  |-  F/_ z { z  |  ph }
31, 2nfdif 3310 . . 3  |-  F/_ z
( _V  \  {
z  |  ph }
)
4 nfab1 2434 . . 3  |-  F/_ z { z  |  -.  ph }
53, 4cleqf 2456 . 2  |-  ( ( _V  \  { z  |  ph } )  =  { z  |  -.  ph }  <->  A. z
( z  e.  ( _V  \  { z  |  ph } )  <-> 
z  e.  { z  |  -.  ph }
) )
6 abid 2284 . . . 4  |-  ( z  e.  { z  | 
ph }  <->  ph )
76notbii 287 . . 3  |-  ( -.  z  e.  { z  |  ph }  <->  -.  ph )
8 compel 27743 . . 3  |-  ( z  e.  ( _V  \  { z  |  ph } )  <->  -.  z  e.  { z  |  ph } )
9 abid 2284 . . 3  |-  ( z  e.  { z  |  -.  ph }  <->  -.  ph )
107, 8, 93bitr4i 268 . 2  |-  ( z  e.  ( _V  \  { z  |  ph } )  <->  z  e.  { z  |  -.  ph } )
115, 10mpgbir 1540 1  |-  ( _V 
\  { z  | 
ph } )  =  { z  |  -.  ph }
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    = wceq 1632    e. wcel 1696   {cab 2282   _Vcvv 2801    \ cdif 3162
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-dif 3168
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