Theorem bj-inrab2 32924

Description: Shorter proof of inrab 3899. (Contributed by BJ, 21-Apr-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-inrab2	|- ( { x e. A | ph } i^i { x e. A | ps } ) = { x e. A | ( ph /\ ps ) }

Proof of Theorem bj-inrab2

Step	Hyp	Ref	Expression
1		bj-inrab 32923	. 2 |- ( { x e. A | ph } i^i { x e. A | ps } ) = { x e. ( A i^i A ) | ( ph /\ ps ) }
2		nfv 1843	. . . 4 |- F/ x T.
3		inidm 3822	. . . . 5 |- ( A i^i A ) = A
4	3	a1i 11	. . . 4 |- ( T. -> ( A i^i A ) = A )
5	2, 4	bj-rabeqd 32916	. . 3 |- ( T. -> { x e. ( A i^i A ) | ( ph /\ ps ) } = { x e. A | ( ph /\ ps ) } )
6	5	trud 1493	. 2 |- { x e. ( A i^i A ) | ( ph /\ ps ) } = { x e. A | ( ph /\ ps ) }
7	1, 6	eqtri 2644	1 |- ( { x e. A | ph } i^i { x e. A | ps } ) = { x e. A | ( ph /\ ps ) }

Syntax hints:   /\ wa 384   = wceq 1483   T. wtru 1484   {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)