Theorem bj-inrab 32923

Description: Generalization of inrab 3899. (Contributed by BJ, 21-Apr-2019.)

Assertion

Ref	Expression
bj-inrab	|- ( { x e. A | ph } i^i { x e. B | ps } ) = { x e. ( A i^i B ) | ( ph /\ ps ) }

Proof of Theorem bj-inrab

Step	Hyp	Ref	Expression
1		an4 865	. . . 4 |- ( ( ( x e. A /\ ph ) /\ ( x e. B /\ ps ) ) <-> ( ( x e. A /\ x e. B ) /\ ( ph /\ ps ) ) )
2		elin 3796	. . . . 5 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
3	2	anbi1i 731	. . . 4 |- ( ( x e. ( A i^i B ) /\ ( ph /\ ps ) ) <-> ( ( x e. A /\ x e. B ) /\ ( ph /\ ps ) ) )
4	1, 3	bitr4i 267	. . 3 |- ( ( ( x e. A /\ ph ) /\ ( x e. B /\ ps ) ) <-> ( x e. ( A i^i B ) /\ ( ph /\ ps ) ) )
5	4	abbii 2739	. 2 |- { x | ( ( x e. A /\ ph ) /\ ( x e. B /\ ps ) ) } = { x | ( x e. ( A i^i B ) /\ ( ph /\ ps ) ) }
6		df-rab 2921	. . . 4 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
7		df-rab 2921	. . . 4 |- { x e. B | ps } = { x | ( x e. B /\ ps ) }
8	6, 7	ineq12i 3812	. . 3 |- ( { x e. A | ph } i^i { x e. B | ps } ) = ( { x | ( x e. A /\ ph ) } i^i { x | ( x e. B /\ ps ) } )
9		inab 3895	. . 3 |- ( { x | ( x e. A /\ ph ) } i^i { x | ( x e. B /\ ps ) } ) = { x | ( ( x e. A /\ ph ) /\ ( x e. B /\ ps ) ) }
10	8, 9	eqtri 2644	. 2 |- ( { x e. A | ph } i^i { x e. B | ps } ) = { x | ( ( x e. A /\ ph ) /\ ( x e. B /\ ps ) ) }
11		df-rab 2921	. 2 |- { x e. ( A i^i B ) | ( ph /\ ps ) } = { x | ( x e. ( A i^i B ) /\ ( ph /\ ps ) ) }
12	5, 10, 11	3eqtr4i 2654	1 |- ( { x e. A | ph } i^i { x e. B | ps } ) = { x e. ( A i^i B ) | ( ph /\ ps ) }

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   {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:  bj-inrab2  32924