Theorem notab 3351

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 2426	. . 3 |- { x e. _V | -. ph } = { x | ( x e. _V /\ -. ph ) }
2		rabab 2710	. . 3 |- { x e. _V | -. ph } = { x | -. ph }
3	1, 2	eqtr3i 2163	. 2 |- { x | ( x e. _V /\ -. ph ) } = { x | -. ph }
4		difab 3350	. . 3 |- ( { x | x e. _V } \ { x | ph } ) = { x | ( x e. _V /\ -. ph ) }
5		abid2 2261	. . . 4 |- { x | x e. _V } = _V
6	5	difeq1i 3195	. . 3 |- ( { x | x e. _V } \ { x | ph } ) = ( _V \ { x | ph } )
7	4, 6	eqtr3i 2163	. 2 |- { x | ( x e. _V /\ -. ph ) } = ( _V \ { x | ph } )
8	3, 7	eqtr3i 2163	1 |- { x | -. ph } = ( _V \ { x | ph } )

Syntax hints:   -. wn 3   /\ wa 103   = wceq 1332   e. wcel 1481   {cab 2126   {crab 2421   _Vcvv 2689   \ cdif 3073

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rab 2426  df-v 2691  df-dif 3078

This theorem is referenced by:  dfif3  3492