Theorem notrab

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

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

Proof

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