Theorem elabgt 2878

Description: Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 2883.) (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 2165	. . . . . . 7 |- ( x e. { x | ph } <-> ph )
2		eleq1 2240	. . . . . . 7 |- ( x = A -> ( x e. { x | ph } <-> A e. { x | ph } ) )
3	1, 2	bitr3id 194	. . . . . 6 |- ( x = A -> ( ph <-> A e. { x | ph } ) )
4	3	bibi1d 233	. . . . 5 |- ( x = A -> ( ( ph <-> ps ) <-> ( A e. { x | ph } <-> ps ) ) )
5	4	biimpd 144	. . . 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 1455	. 2 |- ( A. x ( x = A -> ( ph <-> ps ) ) -> A. x ( x = A -> ( A e. { x | ph } <-> ps ) ) )
8		nfcv 2319	. . . 4 |- F/_ x A
9		nfab1 2321	. . . . . 6 |- F/_ x { x | ph }
10	9	nfel2 2332	. . . . 5 |- F/ x A e. { x | ph }
11		nfv 1528	. . . . 5 |- F/ x ps
12	10, 11	nfbi 1589	. . . 4 |- F/ x ( A e. { x | ph } <-> ps )
13		pm5.5 242	. . . 4 |- ( x = A -> ( ( x = A -> ( A e. { x | ph } <-> ps ) ) <-> ( A e. { x | ph } <-> ps ) ) )
14	8, 12, 13	spcgf 2819	. . 3 |- ( A e. B -> ( A. x ( x = A -> ( A e. { x | ph } <-> ps ) ) -> ( A e. { x | ph } <-> ps ) ) )
15	14	imp 124	. 2 |- ( ( A e. B /\ A. x ( x = A -> ( A e. { x | ph } <-> ps ) ) ) -> ( A e. { x | ph } <-> ps ) )
16	7, 15	sylan2 286	1 |- ( ( A e. B /\ A. x ( x = A -> ( ph <-> ps ) ) ) -> ( A e. { x | ph } <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1351   = wceq 1353   e. wcel 2148   {cab 2163

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739

This theorem is referenced by:  elrab3t  2892