Theorem difrab2 29339

Description: Difference of two restricted class abstractions. Compare with difrab 3901. (Contributed by Thierry Arnoux, 3-Jan-2022.)

Assertion

Ref	Expression
difrab2	|- ( { x e. A | ph } \ { x e. B | ph } ) = { x e. ( A \ B ) | ph }

Proof of Theorem difrab2

Step	Hyp	Ref	Expression
1		nfrab1 3122	. . 3 |- F/_ x { x e. A | ph }
2		nfrab1 3122	. . 3 |- F/_ x { x e. B | ph }
3	1, 2	nfdif 3731	. 2 |- F/_ x ( { x e. A | ph } \ { x e. B | ph } )
4		nfrab1 3122	. 2 |- F/_ x { x e. ( A \ B ) | ph }
5		eldif 3584	. . . . 5 |- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
6	5	anbi1i 731	. . . 4 |- ( ( x e. ( A \ B ) /\ ph ) <-> ( ( x e. A /\ -. x e. B ) /\ ph ) )
7		andi 911	. . . . . . 7 |- ( ( ph /\ ( -. x e. B \/ -. ph ) ) <-> ( ( ph /\ -. x e. B ) \/ ( ph /\ -. ph ) ) )
8		pm3.24 926	. . . . . . . 8 |- -. ( ph /\ -. ph )
9	8	biorfi 422	. . . . . . 7 |- ( ( ph /\ -. x e. B ) <-> ( ( ph /\ -. x e. B ) \/ ( ph /\ -. ph ) ) )
10		ancom 466	. . . . . . 7 |- ( ( ph /\ -. x e. B ) <-> ( -. x e. B /\ ph ) )
11	7, 9, 10	3bitr2i 288	. . . . . 6 |- ( ( ph /\ ( -. x e. B \/ -. ph ) ) <-> ( -. x e. B /\ ph ) )
12	11	anbi2i 730	. . . . 5 |- ( ( x e. A /\ ( ph /\ ( -. x e. B \/ -. ph ) ) ) <-> ( x e. A /\ ( -. x e. B /\ ph ) ) )
13		anass 681	. . . . 5 |- ( ( ( x e. A /\ ph ) /\ ( -. x e. B \/ -. ph ) ) <-> ( x e. A /\ ( ph /\ ( -. x e. B \/ -. ph ) ) ) )
14		anass 681	. . . . 5 |- ( ( ( x e. A /\ -. x e. B ) /\ ph ) <-> ( x e. A /\ ( -. x e. B /\ ph ) ) )
15	12, 13, 14	3bitr4i 292	. . . 4 |- ( ( ( x e. A /\ ph ) /\ ( -. x e. B \/ -. ph ) ) <-> ( ( x e. A /\ -. x e. B ) /\ ph ) )
16	6, 15	bitr4i 267	. . 3 |- ( ( x e. ( A \ B ) /\ ph ) <-> ( ( x e. A /\ ph ) /\ ( -. x e. B \/ -. ph ) ) )
17		rabid 3116	. . 3 |- ( x e. { x e. ( A \ B ) | ph } <-> ( x e. ( A \ B ) /\ ph ) )
18		eldif 3584	. . . 4 |- ( x e. ( { x e. A | ph } \ { x e. B | ph } ) <-> ( x e. { x e. A | ph } /\ -. x e. { x e. B | ph } ) )
19		rabid 3116	. . . . 5 |- ( x e. { x e. A | ph } <-> ( x e. A /\ ph ) )
20		rabid 3116	. . . . . . 7 |- ( x e. { x e. B | ph } <-> ( x e. B /\ ph ) )
21	20	notbii 310	. . . . . 6 |- ( -. x e. { x e. B | ph } <-> -. ( x e. B /\ ph ) )
22		ianor 509	. . . . . 6 |- ( -. ( x e. B /\ ph ) <-> ( -. x e. B \/ -. ph ) )
23	21, 22	bitri 264	. . . . 5 |- ( -. x e. { x e. B | ph } <-> ( -. x e. B \/ -. ph ) )
24	19, 23	anbi12i 733	. . . 4 |- ( ( x e. { x e. A | ph } /\ -. x e. { x e. B | ph } ) <-> ( ( x e. A /\ ph ) /\ ( -. x e. B \/ -. ph ) ) )
25	18, 24	bitri 264	. . 3 |- ( x e. ( { x e. A | ph } \ { x e. B | ph } ) <-> ( ( x e. A /\ ph ) /\ ( -. x e. B \/ -. ph ) ) )
26	16, 17, 25	3bitr4ri 293	. 2 |- ( x e. ( { x e. A | ph } \ { x e. B | ph } ) <-> x e. { x e. ( A \ B ) | ph } )
27	3, 4, 26	eqri 29315	1 |- ( { x e. A | ph } \ { x e. B | ph } ) = { x e. ( A \ B ) | ph }

Syntax hints:   -. wn 3   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   \ cdif 3571

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-dif 3577

This theorem is referenced by:  reprdifc  30705