Theorem bj-inrab3

Description: Generalization of dfrab3ss , which it may shorten. (Contributed by BJ, 21-Apr-2019) (Revised by OpenAI, 7-Jul-2020)

	Ref	Expression
Assertion	bj-inrab3	|- ( A i^i { x e. B | ph } ) = ( { x e. A | ph } i^i B )

Proof

Step	Hyp	Ref	Expression
1		dfrab3	|- { x e. B | ph } = ( B i^i { x | ph } )
2	1	ineq2i	|- ( A i^i { x e. B | ph } ) = ( A i^i ( B i^i { x | ph } ) )
3		dfrab3	|- { x e. A | ph } = ( A i^i { x | ph } )
4	3	ineq2i	|- ( B i^i { x e. A | ph } ) = ( B i^i ( A i^i { x | ph } ) )
5		incom	|- ( { x e. A | ph } i^i B ) = ( B i^i { x e. A | ph } )
6		in12	|- ( A i^i ( B i^i { x | ph } ) ) = ( B i^i ( A i^i { x | ph } ) )
7	4 5 6	3eqtr4i	|- ( { x e. A | ph } i^i B ) = ( A i^i ( B i^i { x | ph } ) )
8	2 7	eqtr4i	|- ( A i^i { x e. B | ph } ) = ( { x e. A | ph } i^i B )