Theorem elrabf 2696

Description: Membership in a restricted class abstraction, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.)

Hypotheses

Ref	Expression
elrabf.1	|- F/_ x A
elrabf.2	|- F/_ x B
elrabf.3	|- F/ x ps
elrabf.4	|- ( x = A -> ( ph <-> ps ) )

Assertion

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

Proof of Theorem elrabf

Step	Hyp	Ref	Expression
1		elex 2566	. 2 |- ( A e. { x e. B | ph } -> A e. _V )
2		elex 2566	. . 3 |- ( A e. B -> A e. _V )
3	2	adantr 261	. 2 |- ( ( A e. B /\ ps ) -> A e. _V )
4		df-rab 2315	. . . 4 |- { x e. B | ph } = { x | ( x e. B /\ ph ) }
5	4	eleq2i 2104	. . 3 |- ( A e. { x e. B | ph } <-> A e. { x | ( x e. B /\ ph ) } )
6		elrabf.1	. . . 4 |- F/_ x A
7		elrabf.2	. . . . . 6 |- F/_ x B
8	6, 7	nfel 2186	. . . . 5 |- F/ x A e. B
9		elrabf.3	. . . . 5 |- F/ x ps
10	8, 9	nfan 1457	. . . 4 |- F/ x ( A e. B /\ ps )
11		eleq1 2100	. . . . 5 |- ( x = A -> ( x e. B <-> A e. B ) )
12		elrabf.4	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
13	11, 12	anbi12d 442	. . . 4 |- ( x = A -> ( ( x e. B /\ ph ) <-> ( A e. B /\ ps ) ) )
14	6, 10, 13	elabgf 2685	. . 3 |- ( A e. _V -> ( A e. { x | ( x e. B /\ ph ) } <-> ( A e. B /\ ps ) ) )
15	5, 14	syl5bb 181	. 2 |- ( A e. _V -> ( A e. { x e. B | ph } <-> ( A e. B /\ ps ) ) )
16	1, 3, 15	pm5.21nii 620	1 |- ( A e. { x e. B | ph } <-> ( A e. B /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   F/wnf 1349   e. wcel 1393   {cab 2026   F/_wnfc 2165   {crab 2310   _Vcvv 2557

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2315  df-v 2559

This theorem is referenced by:  elrab  2698  frind  4089  rabxfrd  4201