Theorem notrab 3353

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 3345	. 2 |- ( { x | x e. A } \ { x | ph } ) = { x | ( x e. A /\ -. ph ) }
2		difin 3313	. . 3 |- ( A \ ( A i^i { x | ph } ) ) = ( A \ { x | ph } )
3		dfrab3 3352	. . . 4 |- { x e. A | ph } = ( A i^i { x | ph } )
4	3	difeq2i 3191	. . 3 |- ( A \ { x e. A | ph } ) = ( A \ ( A i^i { x | ph } ) )
5		abid2 2260	. . . 4 |- { x | x e. A } = A
6	5	difeq1i 3190	. . 3 |- ( { x | x e. A } \ { x | ph } ) = ( A \ { x | ph } )
7	2, 4, 6	3eqtr4i 2170	. 2 |- ( A \ { x e. A | ph } ) = ( { x | x e. A } \ { x | ph } )
8		df-rab 2425	. 2 |- { x e. A | -. ph } = { x | ( x e. A /\ -. ph ) }
9	1, 7, 8	3eqtr4i 2170	1 |- ( A \ { x e. A | ph } ) = { x e. A | -. ph }

Syntax hints:   -. wn 3   /\ wa 103   = wceq 1331   e. wcel 1480   {cab 2125   {crab 2420   \ cdif 3068   i^i cin 3070

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rab 2425  df-v 2688  df-dif 3073  df-in 3077

This theorem is referenced by:  diffitest  6781