Theorem compab 38645

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

Step	Hyp	Ref	Expression
1		nfcv 2764	. . . 4 |- F/_ z _V
2		nfab1 2766	. . . 4 |- F/_ z { z | ph }
3	1, 2	nfdif 3731	. . 3 |- F/_ z ( _V \ { z | ph } )
4		nfab1 2766	. . 3 |- F/_ z { z | -. ph }
5	3, 4	cleqf 2790	. 2 |- ( ( _V \ { z | ph } ) = { z | -. ph } <-> A. z ( z e. ( _V \ { z | ph } ) <-> z e. { z | -. ph } ) )
6		abid 2610	. . . 4 |- ( z e. { z | ph } <-> ph )
7	6	notbii 310	. . 3 |- ( -. z e. { z | ph } <-> -. ph )
8		compel 38641	. . 3 |- ( z e. ( _V \ { z | ph } ) <-> -. z e. { z | ph } )
9		abid 2610	. . 3 |- ( z e. { z | -. ph } <-> -. ph )
10	7, 8, 9	3bitr4i 292	. 2 |- ( z e. ( _V \ { z | ph } ) <-> z e. { z | -. ph } )
11	5, 10	mpgbir 1726	1 |- ( _V \ { z | ph } ) = { z | -. ph }

Syntax hints:   -. wn 3   <-> wb 196   = wceq 1483   e. wcel 1990   {cab 2608   _Vcvv 3200   \ cdif 3571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577

This theorem is referenced by: (None)