Theorem elrab3t 3362

Description: Membership in a restricted class abstraction, using implicit substitution. (Closed theorem version of elrab3 3364.) (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		df-rab 2921	. . 3 |- { x e. B | ph } = { x | ( x e. B /\ ph ) }
2	1	eleq2i 2693	. 2 |- ( A e. { x e. B | ph } <-> A e. { x | ( x e. B /\ ph ) } )
3		id 22	. . 3 |- ( A e. B -> A e. B )
4		nfa1 2028	. . . . 5 |- F/ x A. x ( x = A -> ( ph <-> ps ) )
5		nfv 1843	. . . . 5 |- F/ x A e. B
6	4, 5	nfan 1828	. . . 4 |- F/ x ( A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B )
7		sp 2053	. . . . . 6 |- ( A. x ( x = A -> ( ph <-> ps ) ) -> ( x = A -> ( ph <-> ps ) ) )
8		eleq1 2689	. . . . . . . . . 10 |- ( x = A -> ( x e. B <-> A e. B ) )
9	8	biimparc 504	. . . . . . . . 9 |- ( ( A e. B /\ x = A ) -> x e. B )
10	9	biantrurd 529	. . . . . . . 8 |- ( ( A e. B /\ x = A ) -> ( ph <-> ( x e. B /\ ph ) ) )
11	10	bibi1d 333	. . . . . . 7 |- ( ( A e. B /\ x = A ) -> ( ( ph <-> ps ) <-> ( ( x e. B /\ ph ) <-> ps ) ) )
12	11	pm5.74da 723	. . . . . 6 |- ( A e. B -> ( ( x = A -> ( ph <-> ps ) ) <-> ( x = A -> ( ( x e. B /\ ph ) <-> ps ) ) ) )
13	7, 12	syl5ibcom 235	. . . . 5 |- ( A. x ( x = A -> ( ph <-> ps ) ) -> ( A e. B -> ( x = A -> ( ( x e. B /\ ph ) <-> ps ) ) ) )
14	13	imp 445	. . . 4 |- ( ( A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> ( x = A -> ( ( x e. B /\ ph ) <-> ps ) ) )
15	6, 14	alrimi 2082	. . 3 |- ( ( A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> A. x ( x = A -> ( ( x e. B /\ ph ) <-> ps ) ) )
16		elabgt 3347	. . 3 |- ( ( A e. B /\ A. x ( x = A -> ( ( x e. B /\ ph ) <-> ps ) ) ) -> ( A e. { x | ( x e. B /\ ph ) } <-> ps ) )
17	3, 15, 16	syl2an2 875	. 2 |- ( ( A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> ( A e. { x | ( x e. B /\ ph ) } <-> ps ) )
18	2, 17	syl5bb 272	1 |- ( ( A. x ( x = A -> ( ph <-> ps ) ) /\ A e. B ) -> ( A e. { x e. B | ph } <-> ps ) )

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

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

This theorem is referenced by:  f1oresrab  6395  poimirlem17  33426