Theorem rspc 2650

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 2311	. 2 |- ( A. x e. B ph <-> A. x ( x e. B -> ph ) )
2		nfcv 2178	. . . 4 |- F/_ x A
3		nfv 1421	. . . . 5 |- F/ x A e. B
4		rspc.1	. . . . 5 |- F/ x ps
5	3, 4	nfim 1464	. . . 4 |- F/ x ( A e. B -> ps )
6		eleq1 2100	. . . . 5 |- ( x = A -> ( x e. B <-> A e. B ) )
7		rspc.2	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
8	6, 7	imbi12d 223	. . . 4 |- ( x = A -> ( ( x e. B -> ph ) <-> ( A e. B -> ps ) ) )
9	2, 5, 8	spcgf 2635	. . 3 |- ( A e. B -> ( A. x ( x e. B -> ph ) -> ( A e. B -> ps ) ) )
10	9	pm2.43a 45	. 2 |- ( A e. B -> ( A. x ( x e. B -> ph ) -> ps ) )
11	1, 10	syl5bi 141	1 |- ( A e. B -> ( A. x e. B ph -> ps ) )

Syntax hints:   -> wi 4   <-> wb 98   A.wal 1241   = wceq 1243   F/wnf 1349   e. wcel 1393   A.wral 2306

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-ral 2311  df-v 2559

This theorem is referenced by:  rspcv  2652  rspc2  2661  pofun  4049  fmptcof  5331  fliftfuns  5438  qliftfuns  6190  bj-nntrans  10076