Theorem notrab 3486

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 3478	. 2 |- ( { x | x e. A } \ { x | ph } ) = { x | ( x e. A /\ -. ph ) }
2		difin 3446	. . 3 |- ( A \ ( A i^i { x | ph } ) ) = ( A \ { x | ph } )
3		dfrab3 3485	. . . 4 |- { x e. A | ph } = ( A i^i { x | ph } )
4	3	difeq2i 3324	. . 3 |- ( A \ { x e. A | ph } ) = ( A \ ( A i^i { x | ph } ) )
5		abid2 2353	. . . 4 |- { x | x e. A } = A
6	5	difeq1i 3323	. . 3 |- ( { x | x e. A } \ { x | ph } ) = ( A \ { x | ph } )
7	2, 4, 6	3eqtr4i 2262	. 2 |- ( A \ { x e. A | ph } ) = ( { x | x e. A } \ { x | ph } )
8		df-rab 2520	. 2 |- { x e. A | -. ph } = { x | ( x e. A /\ -. ph ) }
9	1, 7, 8	3eqtr4i 2262	1 |- ( A \ { x e. A | ph } ) = { x e. A | -. ph }

Syntax hints:   -. wn 3   /\ wa 104   = wceq 1398   e. wcel 2202   {cab 2217   {crab 2515   \ cdif 3198   i^i cin 3200

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 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rab 2520  df-v 2805  df-dif 3203  df-in 3207

This theorem is referenced by:  diffitest  7119