Theorem dfrab3ss

Description: Restricted class abstraction with a common superset. (Contributed by Stefan O'Rear, 12-Sep-2015) (Proof shortened by Mario Carneiro, 8-Nov-2015)

	Ref	Expression
Assertion	dfrab3ss	|- ( A C_ B -> { x e. A | ph } = ( A i^i { x e. B | ph } ) )

Proof

Step	Hyp	Ref	Expression
1		df-ss	|- ( A C_ B <-> ( A i^i B ) = A )
2		ineq1	|- ( ( A i^i B ) = A -> ( ( A i^i B ) i^i { x | ph } ) = ( A i^i { x | ph } ) )
3	2	eqcomd	|- ( ( A i^i B ) = A -> ( A i^i { x | ph } ) = ( ( A i^i B ) i^i { x | ph } ) )
4	1 3	sylbi	|- ( A C_ B -> ( A i^i { x | ph } ) = ( ( A i^i B ) i^i { x | ph } ) )
5		dfrab3	|- { x e. A | ph } = ( A i^i { x | ph } )
6		dfrab3	|- { x e. B | ph } = ( B i^i { x | ph } )
7	6	ineq2i	|- ( A i^i { x e. B | ph } ) = ( A i^i ( B i^i { x | ph } ) )
8		inass	|- ( ( A i^i B ) i^i { x | ph } ) = ( A i^i ( B i^i { x | ph } ) )
9	7 8	eqtr4i	|- ( A i^i { x e. B | ph } ) = ( ( A i^i B ) i^i { x | ph } )
10	4 5 9	3eqtr4g	|- ( A C_ B -> { x e. A | ph } = ( A i^i { x e. B | ph } ) )