Theorem elabgt 2684

Description: Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 2688.) (Contributed by NM, 7-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Assertion

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

Distinct variable groups:   x, A   ps, x

Allowed substitution hints:   ph( x)   B( x)

Proof of Theorem elabgt

Step	Hyp	Ref	Expression
1		abid 2028	. . . . . . 7 |- ( x e. { x | ph } <-> ph )
2		eleq1 2100	. . . . . . 7 |- ( x = A -> ( x e. { x | ph } <-> A e. { x | ph } ) )
3	1, 2	syl5bbr 183	. . . . . 6 |- ( x = A -> ( ph <-> A e. { x | ph } ) )
4	3	bibi1d 222	. . . . 5 |- ( x = A -> ( ( ph <-> ps ) <-> ( A e. { x | ph } <-> ps ) ) )
5	4	biimpd 132	. . . 4 |- ( x = A -> ( ( ph <-> ps ) -> ( A e. { x | ph } <-> ps ) ) )
6	5	a2i 11	. . 3 |- ( ( x = A -> ( ph <-> ps ) ) -> ( x = A -> ( A e. { x | ph } <-> ps ) ) )
7	6	alimi 1344	. 2 |- ( A. x ( x = A -> ( ph <-> ps ) ) -> A. x ( x = A -> ( A e. { x | ph } <-> ps ) ) )
8		nfcv 2178	. . . 4 |- F/_ x A
9		nfab1 2180	. . . . . 6 |- F/_ x { x | ph }
10	9	nfel2 2190	. . . . 5 |- F/ x A e. { x | ph }
11		nfv 1421	. . . . 5 |- F/ x ps
12	10, 11	nfbi 1481	. . . 4 |- F/ x ( A e. { x | ph } <-> ps )
13		pm5.5 231	. . . 4 |- ( x = A -> ( ( x = A -> ( A e. { x | ph } <-> ps ) ) <-> ( A e. { x | ph } <-> ps ) ) )
14	8, 12, 13	spcgf 2635	. . 3 |- ( A e. B -> ( A. x ( x = A -> ( A e. { x | ph } <-> ps ) ) -> ( A e. { x | ph } <-> ps ) ) )
15	14	imp 115	. 2 |- ( ( A e. B /\ A. x ( x = A -> ( A e. { x | ph } <-> ps ) ) ) -> ( A e. { x | ph } <-> ps ) )
16	7, 15	sylan2 270	1 |- ( ( A e. B /\ A. x ( x = A -> ( ph <-> ps ) ) ) -> ( A e. { x | ph } <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241   = wceq 1243   e. wcel 1393   {cab 2026

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-v 2559

This theorem is referenced by:  elrab3t  2697