Theorem elabgt 2901

Description: Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 2906.) (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 2181	. . . . . . 7 |- ( x e. { x | ph } <-> ph )
2		eleq1 2256	. . . . . . 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 1466	. 2 |- ( A. x ( x = A -> ( ph <-> ps ) ) -> A. x ( x = A -> ( A e. { x | ph } <-> ps ) ) )
8		nfcv 2336	. . . 4 |- F/_ x A
9		nfab1 2338	. . . . . 6 |- F/_ x { x | ph }
10	9	nfel2 2349	. . . . 5 |- F/ x A e. { x | ph }
11		nfv 1539	. . . . 5 |- F/ x ps
12	10, 11	nfbi 1600	. . . 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 2842	. . 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 1362   = wceq 1364   e. wcel 2164   {cab 2179

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762

This theorem is referenced by:  elrab3t  2915