Theorem bj-inrab3 32925

Description: Generalization of dfrab3ss 3905, which it may shorten. (Contributed by BJ, 21-Apr-2019.) (Revised by OpenAI, 7-Jul-2020.)

Assertion

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

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   ph( x)

Proof of Theorem bj-inrab3

Step	Hyp	Ref	Expression
1		dfrab3 3902	. . 3 |- { x e. B | ph } = ( B i^i { x | ph } )
2	1	ineq2i 3811	. 2 |- ( A i^i { x e. B | ph } ) = ( A i^i ( B i^i { x | ph } ) )
3		dfrab3 3902	. . . 4 |- { x e. A | ph } = ( A i^i { x | ph } )
4	3	ineq2i 3811	. . 3 |- ( B i^i { x e. A | ph } ) = ( B i^i ( A i^i { x | ph } ) )
5		incom 3805	. . 3 |- ( { x e. A | ph } i^i B ) = ( B i^i { x e. A | ph } )
6		in12 3824	. . 3 |- ( A i^i ( B i^i { x | ph } ) ) = ( B i^i ( A i^i { x | ph } ) )
7	4, 5, 6	3eqtr4i 2654	. 2 |- ( { x e. A | ph } i^i B ) = ( A i^i ( B i^i { x | ph } ) )
8	2, 7	eqtr4i 2647	1 |- ( A i^i { x e. B | ph } ) = ( { x e. A | ph } i^i B )

Syntax hints:   = wceq 1483   {cab 2608   {crab 2916   i^i cin 3573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-in 3581

This theorem is referenced by: (None)