Theorem notab 3474

Description: A class builder defined by a negation. (Contributed by FL, 18-Sep-2010.)

Assertion

Ref	Expression
notab	|- { x | -. ph } = ( _V \ { x | ph } )

Proof of Theorem notab

Step	Hyp	Ref	Expression
1		df-rab 2517	. . 3 |- { x e. _V | -. ph } = { x | ( x e. _V /\ -. ph ) }
2		rabab 2821	. . 3 |- { x e. _V | -. ph } = { x | -. ph }
3	1, 2	eqtr3i 2252	. 2 |- { x | ( x e. _V /\ -. ph ) } = { x | -. ph }
4		difab 3473	. . 3 |- ( { x | x e. _V } \ { x | ph } ) = { x | ( x e. _V /\ -. ph ) }
5		abid2 2350	. . . 4 |- { x | x e. _V } = _V
6	5	difeq1i 3318	. . 3 |- ( { x | x e. _V } \ { x | ph } ) = ( _V \ { x | ph } )
7	4, 6	eqtr3i 2252	. 2 |- { x | ( x e. _V /\ -. ph ) } = ( _V \ { x | ph } )
8	3, 7	eqtr3i 2252	1 |- { x | -. ph } = ( _V \ { x | ph } )

Syntax hints:   -. wn 3   /\ wa 104   = wceq 1395   e. wcel 2200   {cab 2215   {crab 2512   _Vcvv 2799   \ cdif 3194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rab 2517  df-v 2801  df-dif 3199

This theorem is referenced by:  dfif3  3616