Theorem elrabf 2889

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 2746	. 2 |- ( A e. { x e. B | ph } -> A e. _V )
2		elex 2746	. . 3 |- ( A e. B -> A e. _V )
3	2	adantr 276	. 2 |- ( ( A e. B /\ ps ) -> A e. _V )
4		df-rab 2462	. . . 4 |- { x e. B | ph } = { x | ( x e. B /\ ph ) }
5	4	eleq2i 2242	. . 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 2326	. . . . 5 |- F/ x A e. B
9		elrabf.3	. . . . 5 |- F/ x ps
10	8, 9	nfan 1563	. . . 4 |- F/ x ( A e. B /\ ps )
11		eleq1 2238	. . . . 5 |- ( x = A -> ( x e. B <-> A e. B ) )
12		elrabf.4	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
13	11, 12	anbi12d 473	. . . 4 |- ( x = A -> ( ( x e. B /\ ph ) <-> ( A e. B /\ ps ) ) )
14	6, 10, 13	elabgf 2877	. . 3 |- ( A e. _V -> ( A e. { x | ( x e. B /\ ph ) } <-> ( A e. B /\ ps ) ) )
15	5, 14	bitrid 192	. 2 |- ( A e. _V -> ( A e. { x e. B | ph } <-> ( A e. B /\ ps ) ) )
16	1, 3, 15	pm5.21nii 704	1 |- ( A e. { x e. B | ph } <-> ( A e. B /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   F/wnf 1458   e. wcel 2146   {cab 2161   F/_wnfc 2304   {crab 2457   _Vcvv 2735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-rab 2462  df-v 2737

This theorem is referenced by:  elrab  2891  frind  4346  rabxfrd  4463  infssuzcldc  11917  nnwosdc  12005