Theorem elrabf 2994

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/_xA
elrabf.2	|- F/_xB
elrabf.3	|- F/xps
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 2868	. 2 |- (A e. {x e. B | ph} -> A e. _V)
2		elex 2868	. . 3 |- (A e. B -> A e. _V)
3	2	adantr 451	. 2 |- ((A e. B /\ ps) -> A e. _V)
4		df-rab 2624	. . . 4 |- {x e. B | ph} = {x | (x e. B /\ ph)}
5	4	eleq2i 2417	. . 3 |- (A e. {x e. B | ph} <-> A e. {x | (x e. B /\ ph)})
6		elrabf.1	. . . 4 |- F/_xA
7		elrabf.2	. . . . . 6 |- F/_xB
8	6, 7	nfel 2498	. . . . 5 |- F/x A e. B
9		elrabf.3	. . . . 5 |- F/xps
10	8, 9	nfan 1824	. . . 4 |- F/x(A e. B /\ ps)
11		eleq1 2413	. . . . 5 |- (x = A -> (x e. B <-> A e. B))
12		elrabf.4	. . . . 5 |- (x = A -> (ph <-> ps))
13	11, 12	anbi12d 691	. . . 4 |- (x = A -> ((x e. B /\ ph) <-> (A e. B /\ ps)))
14	6, 10, 13	elabgf 2984	. . 3 |- (A e. _V -> (A e. {x | (x e. B /\ ph)} <-> (A e. B /\ ps)))
15	5, 14	syl5bb 248	. 2 |- (A e. _V -> (A e. {x e. B | ph} <-> (A e. B /\ ps)))
16	1, 3, 15	pm5.21nii 342	1 |- (A e. {x e. B | ph} <-> (A e. B /\ ps))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   F/wnf 1544   = wceq 1642   e. wcel 1710  {cab 2339   F/_wnfc 2477  {crab 2619  _Vcvv 2860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rab 2624  df-v 2862

This theorem is referenced by:  elrab  2995