Theorem elrabf 2957

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 2811	. 2 |- ( A e. { x e. B | ph } -> A e. _V )
2		elex 2811	. . 3 |- ( A e. B -> A e. _V )
3	2	adantr 276	. 2 |- ( ( A e. B /\ ps ) -> A e. _V )
4		df-rab 2517	. . . 4 |- { x e. B | ph } = { x | ( x e. B /\ ph ) }
5	4	eleq2i 2296	. . 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 2381	. . . . 5 |- F/ x A e. B
9		elrabf.3	. . . . 5 |- F/ x ps
10	8, 9	nfan 1611	. . . 4 |- F/ x ( A e. B /\ ps )
11		eleq1 2292	. . . . 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 2945	. . 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 709	1 |- ( A e. { x e. B | ph } <-> ( A e. B /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   F/wnf 1506   e. wcel 2200   {cab 2215   F/_wnfc 2359   {crab 2512   _Vcvv 2799

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rab 2517  df-v 2801

This theorem is referenced by:  elrab  2959  invdisjrab  4077  frind  4443  rabxfrd  4560  infssuzcldc  10455  nnwosdc  12560