Theorem notrab 3259

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 3251	. 2 |- ( { x | x e. A } \ { x | ph } ) = { x | ( x e. A /\ -. ph ) }
2		difin 3219	. . 3 |- ( A \ ( A i^i { x | ph } ) ) = ( A \ { x | ph } )
3		dfrab3 3258	. . . 4 |- { x e. A | ph } = ( A i^i { x | ph } )
4	3	difeq2i 3099	. . 3 |- ( A \ { x e. A | ph } ) = ( A \ ( A i^i { x | ph } ) )
5		abid2 2203	. . . 4 |- { x | x e. A } = A
6	5	difeq1i 3098	. . 3 |- ( { x | x e. A } \ { x | ph } ) = ( A \ { x | ph } )
7	2, 4, 6	3eqtr4i 2113	. 2 |- ( A \ { x e. A | ph } ) = ( { x | x e. A } \ { x | ph } )
8		df-rab 2362	. 2 |- { x e. A | -. ph } = { x | ( x e. A /\ -. ph ) }
9	1, 7, 8	3eqtr4i 2113	1 |- ( A \ { x e. A | ph } ) = { x e. A | -. ph }

Syntax hints:   -. wn 3   /\ wa 102   = wceq 1285   e. wcel 1434   {cab 2069   {crab 2357   \ cdif 2981   i^i cin 2983

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rab 2362  df-v 2614  df-dif 2986  df-in 2990

This theorem is referenced by:  diffitest  6533