Theorem elabgt 2771

Description: Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 2775.) (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 2083	. . . . . . 7 |- ( x e. { x | ph } <-> ph )
2		eleq1 2157	. . . . . . 7 |- ( x = A -> ( x e. { x | ph } <-> A e. { x | ph } ) )
3	1, 2	syl5bbr 193	. . . . . 6 |- ( x = A -> ( ph <-> A e. { x | ph } ) )
4	3	bibi1d 232	. . . . 5 |- ( x = A -> ( ( ph <-> ps ) <-> ( A e. { x | ph } <-> ps ) ) )
5	4	biimpd 143	. . . 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 1396	. 2 |- ( A. x ( x = A -> ( ph <-> ps ) ) -> A. x ( x = A -> ( A e. { x | ph } <-> ps ) ) )
8		nfcv 2235	. . . 4 |- F/_ x A
9		nfab1 2237	. . . . . 6 |- F/_ x { x | ph }
10	9	nfel2 2248	. . . . 5 |- F/ x A e. { x | ph }
11		nfv 1473	. . . . 5 |- F/ x ps
12	10, 11	nfbi 1533	. . . 4 |- F/ x ( A e. { x | ph } <-> ps )
13		pm5.5 241	. . . 4 |- ( x = A -> ( ( x = A -> ( A e. { x | ph } <-> ps ) ) <-> ( A e. { x | ph } <-> ps ) ) )
14	8, 12, 13	spcgf 2715	. . 3 |- ( A e. B -> ( A. x ( x = A -> ( A e. { x | ph } <-> ps ) ) -> ( A e. { x | ph } <-> ps ) ) )
15	14	imp 123	. 2 |- ( ( A e. B /\ A. x ( x = A -> ( A e. { x | ph } <-> ps ) ) ) -> ( A e. { x | ph } <-> ps ) )
16	7, 15	sylan2 281	1 |- ( ( A e. B /\ A. x ( x = A -> ( ph <-> ps ) ) ) -> ( A e. { x | ph } <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1294   = wceq 1296   e. wcel 1445   {cab 2081

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635

This theorem is referenced by:  elrab3t  2784