Theorem rababg 37879

Description: Condition when restricted class is equal to unrestricted class. (Contributed by RP, 13-Aug-2020.)

Assertion

Ref	Expression
rababg	|- ( A. x ( ph -> x e. A ) <-> { x e. A | ph } = { x | ph } )

Proof of Theorem rababg

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ancrb 573	. . 3 |- ( ( ph -> x e. A ) <-> ( ph -> ( x e. A /\ ph ) ) )
2	1	albii 1747	. 2 |- ( A. x ( ph -> x e. A ) <-> A. x ( ph -> ( x e. A /\ ph ) ) )
3		nfv 1843	. . 3 |- F/ y ( ph -> ( x e. A /\ ph ) )
4		nfsab1 2612	. . . 4 |- F/ x y e. { x | ph }
5		nfrab1 3122	. . . . 5 |- F/_ x { x e. A | ph }
6	5	nfcri 2758	. . . 4 |- F/ x y e. { x e. A | ph }
7	4, 6	nfim 1825	. . 3 |- F/ x ( y e. { x | ph } -> y e. { x e. A | ph } )
8		abid 2610	. . . . 5 |- ( x e. { x | ph } <-> ph )
9		eleq1 2689	. . . . 5 |- ( x = y -> ( x e. { x | ph } <-> y e. { x | ph } ) )
10	8, 9	syl5bbr 274	. . . 4 |- ( x = y -> ( ph <-> y e. { x | ph } ) )
11		rabid 3116	. . . . 5 |- ( x e. { x e. A | ph } <-> ( x e. A /\ ph ) )
12		eleq1 2689	. . . . 5 |- ( x = y -> ( x e. { x e. A | ph } <-> y e. { x e. A | ph } ) )
13	11, 12	syl5bbr 274	. . . 4 |- ( x = y -> ( ( x e. A /\ ph ) <-> y e. { x e. A | ph } ) )
14	10, 13	imbi12d 334	. . 3 |- ( x = y -> ( ( ph -> ( x e. A /\ ph ) ) <-> ( y e. { x | ph } -> y e. { x e. A | ph } ) ) )
15	3, 7, 14	cbval 2271	. 2 |- ( A. x ( ph -> ( x e. A /\ ph ) ) <-> A. y ( y e. { x | ph } -> y e. { x e. A | ph } ) )
16		eqss 3618	. . 3 |- ( { x e. A | ph } = { x | ph } <-> ( { x e. A | ph } C_ { x | ph } /\ { x | ph } C_ { x e. A | ph } ) )
17		rabssab 3690	. . . 4 |- { x e. A | ph } C_ { x | ph }
18	17	biantrur 527	. . 3 |- ( { x | ph } C_ { x e. A | ph } <-> ( { x e. A | ph } C_ { x | ph } /\ { x | ph } C_ { x e. A | ph } ) )
19		dfss2 3591	. . 3 |- ( { x | ph } C_ { x e. A | ph } <-> A. y ( y e. { x | ph } -> y e. { x e. A | ph } ) )
20	16, 18, 19	3bitr2ri 289	. 2 |- ( A. y ( y e. { x | ph } -> y e. { x e. A | ph } ) <-> { x e. A | ph } = { x | ph } )
21	2, 15, 20	3bitri 286	1 |- ( A. x ( ph -> x e. A ) <-> { x e. A | ph } = { x | ph } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   {cab 2608   {crab 2916   C_ wss 3574

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-in 3581  df-ss 3588

This theorem is referenced by: (None)