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Theorem compab 29837
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 2613 . . . 4  |-  F/_ z _V
2 nfab1 2615 . . . 4  |-  F/_ z { z  |  ph }
31, 2nfdif 3577 . . 3  |-  F/_ z
( _V  \  {
z  |  ph }
)
4 nfab1 2615 . . 3  |-  F/_ z { z  |  -.  ph }
53, 4cleqf 2639 . 2  |-  ( ( _V  \  { z  |  ph } )  =  { z  |  -.  ph }  <->  A. z
( z  e.  ( _V  \  { z  |  ph } )  <-> 
z  e.  { z  |  -.  ph }
) )
6 abid 2438 . . . 4  |-  ( z  e.  { z  | 
ph }  <->  ph )
76notbii 296 . . 3  |-  ( -.  z  e.  { z  |  ph }  <->  -.  ph )
8 compel 29834 . . 3  |-  ( z  e.  ( _V  \  { z  |  ph } )  <->  -.  z  e.  { z  |  ph } )
9 abid 2438 . . 3  |-  ( z  e.  { z  |  -.  ph }  <->  -.  ph )
107, 8, 93bitr4i 277 . 2  |-  ( z  e.  ( _V  \  { z  |  ph } )  <->  z  e.  { z  |  -.  ph } )
115, 10mpgbir 1596 1  |-  ( _V 
\  { z  | 
ph } )  =  { z  |  -.  ph }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    = wceq 1370    e. wcel 1758   {cab 2436   _Vcvv 3070    \ cdif 3425
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-rab 2804  df-v 3072  df-dif 3431
This theorem is referenced by: (None)
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