Theorem elrab3t 2748

Description: Membership in a restricted class abstraction, using implicit substitution. (Closed theorem version of elrab3 2750.) (Contributed by Thierry Arnoux, 31-Aug-2017.)

Assertion

Ref	Expression
elrab3t	|- ( ( A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> ( A e. { x e. B | ph } <-> ps ) )

Distinct variable groups:   x, A   x, B   ps, x

Allowed substitution hint:   ph( x)

Proof of Theorem elrab3t

Step	Hyp	Ref	Expression
1		simpr 108	. . 3 |- ( ( A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> A e. B )
2		nfa1 1474	. . . . 5 |- F/ x A. x ( x = A -> ( ph <-> ps ) )
3		nfv 1461	. . . . 5 |- F/ x A e. B
4	2, 3	nfan 1497	. . . 4 |- F/ x ( A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B )
5		simpl 107	. . . . . 6 |- ( ( A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> A. x ( x = A -> ( ph <-> ps ) ) )
6	5	19.21bi 1490	. . . . 5 |- ( ( A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> ( x = A -> ( ph <-> ps ) ) )
7		eleq1 2141	. . . . . . . . . 10 |- ( x = A -> ( x e. B <-> A e. B ) )
8	7	biimparc 293	. . . . . . . . 9 |- ( ( A e. B /\ x = A ) -> x e. B )
9	8	biantrurd 299	. . . . . . . 8 |- ( ( A e. B /\ x = A ) -> ( ph <-> ( x e. B /\ ph ) ) )
10	9	bibi1d 231	. . . . . . 7 |- ( ( A e. B /\ x = A ) -> ( ( ph <-> ps ) <-> ( ( x e. B /\ ph ) <-> ps ) ) )
11	10	pm5.74da 431	. . . . . 6 |- ( A e. B -> ( ( x = A -> ( ph <-> ps ) ) <-> ( x = A -> ( ( x e. B /\ ph ) <-> ps ) ) ) )
12	11	adantl 271	. . . . 5 |- ( ( A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> ( ( x = A -> ( ph <-> ps ) ) <-> ( x = A -> ( ( x e. B /\ ph ) <-> ps ) ) ) )
13	6, 12	mpbid 145	. . . 4 |- ( ( A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> ( x = A -> ( ( x e. B /\ ph ) <-> ps ) ) )
14	4, 13	alrimi 1455	. . 3 |- ( ( A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> A. x ( x = A -> ( ( x e. B /\ ph ) <-> ps ) ) )
15		elabgt 2735	. . 3 |- ( ( A e. B /\ A. x ( x = A -> ( ( x e. B /\ ph ) <-> ps ) ) ) -> ( A e. { x | ( x e. B /\ ph ) } <-> ps ) )
16	1, 14, 15	syl2anc 403	. 2 |- ( ( A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> ( A e. { x | ( x e. B /\ ph ) } <-> ps ) )
17		df-rab 2357	. . . 4 |- { x e. B | ph } = { x | ( x e. B /\ ph ) }
18	17	eleq2i 2145	. . 3 |- ( A e. { x e. B | ph } <-> A e. { x | ( x e. B /\ ph ) } )
19	18	bibi1i 226	. 2 |- ( ( A e. { x e. B | ph } <-> ps ) <-> ( A e. { x | ( x e. B /\ ph ) } <-> ps ) )
20	16, 19	sylibr 132	1 |- ( ( A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> ( A e. { x e. B | ph } <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1282   = wceq 1284   e. wcel 1433   {cab 2067   {crab 2352

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rab 2357  df-v 2603

This theorem is referenced by:  f1oresrab  5350