Theorem notab 3392

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 2453	. . 3 |- { x e. _V | -. ph } = { x | ( x e. _V /\ -. ph ) }
2		rabab 2747	. . 3 |- { x e. _V | -. ph } = { x | -. ph }
3	1, 2	eqtr3i 2188	. 2 |- { x | ( x e. _V /\ -. ph ) } = { x | -. ph }
4		difab 3391	. . 3 |- ( { x | x e. _V } \ { x | ph } ) = { x | ( x e. _V /\ -. ph ) }
5		abid2 2287	. . . 4 |- { x | x e. _V } = _V
6	5	difeq1i 3236	. . 3 |- ( { x | x e. _V } \ { x | ph } ) = ( _V \ { x | ph } )
7	4, 6	eqtr3i 2188	. 2 |- { x | ( x e. _V /\ -. ph ) } = ( _V \ { x | ph } )
8	3, 7	eqtr3i 2188	1 |- { x | -. ph } = ( _V \ { x | ph } )

Syntax hints:   -. wn 3   /\ wa 103   = wceq 1343   e. wcel 2136   {cab 2151   {crab 2448   _Vcvv 2726   \ cdif 3113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rab 2453  df-v 2728  df-dif 3118

This theorem is referenced by:  dfif3  3533