Theorem rspc 2695

Description: Restricted specialization, using implicit substitution. (Contributed by NM, 19-Apr-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)

Hypotheses

Ref	Expression
rspc.1	|- F/ x ps
rspc.2	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
rspc	|- ( A e. B -> ( A. x e. B ph -> ps ) )

Distinct variable groups:   x, A   x, B

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

Proof of Theorem rspc

Step	Hyp	Ref	Expression
1		df-ral 2353	. 2 |- ( A. x e. B ph <-> A. x ( x e. B -> ph ) )
2		nfcv 2219	. . . 4 |- F/_ x A
3		nfv 1461	. . . . 5 |- F/ x A e. B
4		rspc.1	. . . . 5 |- F/ x ps
5	3, 4	nfim 1504	. . . 4 |- F/ x ( A e. B -> ps )
6		eleq1 2141	. . . . 5 |- ( x = A -> ( x e. B <-> A e. B ) )
7		rspc.2	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
8	6, 7	imbi12d 232	. . . 4 |- ( x = A -> ( ( x e. B -> ph ) <-> ( A e. B -> ps ) ) )
9	2, 5, 8	spcgf 2680	. . 3 |- ( A e. B -> ( A. x ( x e. B -> ph ) -> ( A e. B -> ps ) ) )
10	9	pm2.43a 50	. 2 |- ( A e. B -> ( A. x ( x e. B -> ph ) -> ps ) )
11	1, 10	syl5bi 150	1 |- ( A e. B -> ( A. x e. B ph -> ps ) )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1282   = wceq 1284   F/wnf 1389   e. wcel 1433   A.wral 2348

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603

This theorem is referenced by:  rspcv  2697  rspc2  2711  pofun  4067  fmptcof  5352  fliftfuns  5458  qliftfuns  6213  lble  8025  exfzdc  9249  uzsinds  9428  zsupcllemstep  10341  infssuzex  10345  bezoutlemmain  10387  bj-nntrans  10746