Theorem notab 3491

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 2529	. . 3 |- { x e. _V | -. ph } = { x | ( x e. _V /\ -. ph ) }
2		rabab 2835	. . 3 |- { x e. _V | -. ph } = { x | -. ph }
3	1, 2	eqtr3i 2255	. 2 |- { x | ( x e. _V /\ -. ph ) } = { x | -. ph }
4		difab 3490	. . 3 |- ( { x | x e. _V } \ { x | ph } ) = { x | ( x e. _V /\ -. ph ) }
5		abid2 2355	. . . 4 |- { x | x e. _V } = _V
6	5	difeq1i 3333	. . 3 |- ( { x | x e. _V } \ { x | ph } ) = ( _V \ { x | ph } )
7	4, 6	eqtr3i 2255	. 2 |- { x | ( x e. _V /\ -. ph ) } = ( _V \ { x | ph } )
8	3, 7	eqtr3i 2255	1 |- { x | -. ph } = ( _V \ { x | ph } )

Syntax hints:   -. wn 3   /\ wa 104   = wceq 1398   e. wcel 2203   {cab 2218   {crab 2524   _Vcvv 2813   \ cdif 3208

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rab 2529  df-v 2815  df-dif 3213

This theorem is referenced by:  dfif3  3636