Theorem rspct 2694

Description: A closed version of rspc 2695. (Contributed by Andrew Salmon, 6-Jun-2011.)

Hypothesis

Ref	Expression
rspct.1	|- F/ x ps

Assertion

Ref	Expression
rspct	|- ( A. x ( x = A -> ( ph <-> ps ) ) -> ( 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 rspct

Step	Hyp	Ref	Expression
1		df-ral 2353	. . . 4 |- ( A. x e. B ph <-> A. x ( x e. B -> ph ) )
2		eleq1 2141	. . . . . . . . . 10 |- ( x = A -> ( x e. B <-> A e. B ) )
3	2	adantr 270	. . . . . . . . 9 |- ( ( x = A /\ ( ph <-> ps ) ) -> ( x e. B <-> A e. B ) )
4		simpr 108	. . . . . . . . 9 |- ( ( x = A /\ ( ph <-> ps ) ) -> ( ph <-> ps ) )
5	3, 4	imbi12d 232	. . . . . . . 8 |- ( ( x = A /\ ( ph <-> ps ) ) -> ( ( x e. B -> ph ) <-> ( A e. B -> ps ) ) )
6	5	ex 113	. . . . . . 7 |- ( x = A -> ( ( ph <-> ps ) -> ( ( x e. B -> ph ) <-> ( A e. B -> ps ) ) ) )
7	6	a2i 11	. . . . . 6 |- ( ( x = A -> ( ph <-> ps ) ) -> ( x = A -> ( ( x e. B -> ph ) <-> ( A e. B -> ps ) ) ) )
8	7	alimi 1384	. . . . 5 |- ( A. x ( x = A -> ( ph <-> ps ) ) -> A. x ( x = A -> ( ( x e. B -> ph ) <-> ( A e. B -> ps ) ) ) )
9		nfv 1461	. . . . . . 7 |- F/ x A e. B
10		rspct.1	. . . . . . 7 |- F/ x ps
11	9, 10	nfim 1504	. . . . . 6 |- F/ x ( A e. B -> ps )
12		nfcv 2219	. . . . . 6 |- F/_ x A
13	11, 12	spcgft 2675	. . . . 5 |- ( A. x ( x = A -> ( ( x e. B -> ph ) <-> ( A e. B -> ps ) ) ) -> ( A e. B -> ( A. x ( x e. B -> ph ) -> ( A e. B -> ps ) ) ) )
14	8, 13	syl 14	. . . 4 |- ( A. x ( x = A -> ( ph <-> ps ) ) -> ( A e. B -> ( A. x ( x e. B -> ph ) -> ( A e. B -> ps ) ) ) )
15	1, 14	syl7bi 163	. . 3 |- ( A. x ( x = A -> ( ph <-> ps ) ) -> ( A e. B -> ( A. x e. B ph -> ( A e. B -> ps ) ) ) )
16	15	com34 82	. 2 |- ( A. x ( x = A -> ( ph <-> ps ) ) -> ( A e. B -> ( A e. B -> ( A. x e. B ph -> ps ) ) ) )
17	16	pm2.43d 49	1 |- ( A. x ( x = A -> ( ph <-> ps ) ) -> ( A e. B -> ( A. x e. B ph -> ps ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> 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: (None)