Theorem notrab 3427

Description: Complementation of restricted class abstractions. (Contributed by Mario Carneiro, 3-Sep-2015.)

Assertion

Ref	Expression
notrab	|- ( A \ { x e. A | ph } ) = { x e. A | -. ph }

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem notrab

Step	Hyp	Ref	Expression
1		difab 3419	. 2 |- ( { x | x e. A } \ { x | ph } ) = { x | ( x e. A /\ -. ph ) }
2		difin 3387	. . 3 |- ( A \ ( A i^i { x | ph } ) ) = ( A \ { x | ph } )
3		dfrab3 3426	. . . 4 |- { x e. A | ph } = ( A i^i { x | ph } )
4	3	difeq2i 3265	. . 3 |- ( A \ { x e. A | ph } ) = ( A \ ( A i^i { x | ph } ) )
5		abid2 2310	. . . 4 |- { x | x e. A } = A
6	5	difeq1i 3264	. . 3 |- ( { x | x e. A } \ { x | ph } ) = ( A \ { x | ph } )
7	2, 4, 6	3eqtr4i 2220	. 2 |- ( A \ { x e. A | ph } ) = ( { x | x e. A } \ { x | ph } )
8		df-rab 2477	. 2 |- { x e. A | -. ph } = { x | ( x e. A /\ -. ph ) }
9	1, 7, 8	3eqtr4i 2220	1 |- ( A \ { x e. A | ph } ) = { x e. A | -. ph }

Syntax hints:   -. wn 3   /\ wa 104   = wceq 1364   e. wcel 2160   {cab 2175   {crab 2472   \ cdif 3141   i^i cin 3143

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-ral 2473  df-rab 2477  df-v 2754  df-dif 3146  df-in 3150

This theorem is referenced by:  diffitest  6916