Theorem difin2

Description: Represent a class difference as an intersection with a larger difference. (Contributed by Jeff Madsen, 2-Sep-2009)

	Ref	Expression
Assertion	difin2	|- ( A C_ C -> ( A \ B ) = ( ( C \ B ) i^i A ) )

Proof

Step	Hyp	Ref	Expression
1		ssel	|- ( A C_ C -> ( x e. A -> x e. C ) )
2	1	pm4.71d	|- ( A C_ C -> ( x e. A <-> ( x e. A /\ x e. C ) ) )
3	2	anbi1d	|- ( A C_ C -> ( ( x e. A /\ -. x e. B ) <-> ( ( x e. A /\ x e. C ) /\ -. x e. B ) ) )
4		eldif	|- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
5		elin	|- ( x e. ( ( C \ B ) i^i A ) <-> ( x e. ( C \ B ) /\ x e. A ) )
6		eldif	|- ( x e. ( C \ B ) <-> ( x e. C /\ -. x e. B ) )
7	6	anbi1i	|- ( ( x e. ( C \ B ) /\ x e. A ) <-> ( ( x e. C /\ -. x e. B ) /\ x e. A ) )
8		ancom	|- ( ( ( x e. C /\ -. x e. B ) /\ x e. A ) <-> ( x e. A /\ ( x e. C /\ -. x e. B ) ) )
9		anass	|- ( ( ( x e. A /\ x e. C ) /\ -. x e. B ) <-> ( x e. A /\ ( x e. C /\ -. x e. B ) ) )
10	8 9	bitr4i	|- ( ( ( x e. C /\ -. x e. B ) /\ x e. A ) <-> ( ( x e. A /\ x e. C ) /\ -. x e. B ) )
11	5 7 10	3bitri	|- ( x e. ( ( C \ B ) i^i A ) <-> ( ( x e. A /\ x e. C ) /\ -. x e. B ) )
12	3 4 11	3bitr4g	|- ( A C_ C -> ( x e. ( A \ B ) <-> x e. ( ( C \ B ) i^i A ) ) )
13	12	eqrdv	|- ( A C_ C -> ( A \ B ) = ( ( C \ B ) i^i A ) )