Theorem notab 3425

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 2477	. . 3 |- { x e. _V | -. ph } = { x | ( x e. _V /\ -. ph ) }
2		rabab 2777	. . 3 |- { x e. _V | -. ph } = { x | -. ph }
3	1, 2	eqtr3i 2212	. 2 |- { x | ( x e. _V /\ -. ph ) } = { x | -. ph }
4		difab 3424	. . 3 |- ( { x | x e. _V } \ { x | ph } ) = { x | ( x e. _V /\ -. ph ) }
5		abid2 2310	. . . 4 |- { x | x e. _V } = _V
6	5	difeq1i 3269	. . 3 |- ( { x | x e. _V } \ { x | ph } ) = ( _V \ { x | ph } )
7	4, 6	eqtr3i 2212	. 2 |- { x | ( x e. _V /\ -. ph ) } = ( _V \ { x | ph } )
8	3, 7	eqtr3i 2212	1 |- { x | -. ph } = ( _V \ { x | ph } )

Syntax hints:   -. wn 3   /\ wa 104   = wceq 1364   e. wcel 2160   {cab 2175   {crab 2472   _Vcvv 2756   \ cdif 3146

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rab 2477  df-v 2758  df-dif 3151

This theorem is referenced by:  dfif3  3566