Theorem elrabf 2927

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 2783	. 2 |- ( A e. { x e. B | ph } -> A e. _V )
2		elex 2783	. . 3 |- ( A e. B -> A e. _V )
3	2	adantr 276	. 2 |- ( ( A e. B /\ ps ) -> A e. _V )
4		df-rab 2493	. . . 4 |- { x e. B | ph } = { x | ( x e. B /\ ph ) }
5	4	eleq2i 2272	. . 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 2357	. . . . 5 |- F/ x A e. B
9		elrabf.3	. . . . 5 |- F/ x ps
10	8, 9	nfan 1588	. . . 4 |- F/ x ( A e. B /\ ps )
11		eleq1 2268	. . . . 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 2915	. . 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 706	1 |- ( A e. { x e. B | ph } <-> ( A e. B /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   F/wnf 1483   e. wcel 2176   {cab 2191   F/_wnfc 2335   {crab 2488   _Vcvv 2772

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rab 2493  df-v 2774

This theorem is referenced by:  elrab  2929  frind  4399  rabxfrd  4516  infssuzcldc  10378  nnwosdc  12360