Theorem rspct 2809

Description: A closed version of rspc 2810. (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 2440	. . . 4 |- ( A. x e. B ph <-> A. x ( x e. B -> ph ) )
2		eleq1 2220	. . . . . . . . . 10 |- ( x = A -> ( x e. B <-> A e. B ) )
3	2	adantr 274	. . . . . . . . 9 |- ( ( x = A /\ ( ph <-> ps ) ) -> ( x e. B <-> A e. B ) )
4		simpr 109	. . . . . . . . 9 |- ( ( x = A /\ ( ph <-> ps ) ) -> ( ph <-> ps ) )
5	3, 4	imbi12d 233	. . . . . . . 8 |- ( ( x = A /\ ( ph <-> ps ) ) -> ( ( x e. B -> ph ) <-> ( A e. B -> ps ) ) )
6	5	ex 114	. . . . . . 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 1435	. . . . 5 |- ( A. x ( x = A -> ( ph <-> ps ) ) -> A. x ( x = A -> ( ( x e. B -> ph ) <-> ( A e. B -> ps ) ) ) )
9		nfv 1508	. . . . . . 7 |- F/ x A e. B
10		rspct.1	. . . . . . 7 |- F/ x ps
11	9, 10	nfim 1552	. . . . . 6 |- F/ x ( A e. B -> ps )
12		nfcv 2299	. . . . . 6 |- F/_ x A
13	11, 12	spcgft 2789	. . . . 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 164	. . 3 |- ( A. x ( x = A -> ( ph <-> ps ) ) -> ( A e. B -> ( A. x e. B ph -> ( A e. B -> ps ) ) ) )
16	15	com34 83	. 2 |- ( A. x ( x = A -> ( ph <-> ps ) ) -> ( A e. B -> ( A e. B -> ( A. x e. B ph -> ps ) ) ) )
17	16	pm2.43d 50	1 |- ( A. x ( x = A -> ( ph <-> ps ) ) -> ( A e. B -> ( A. x e. B ph -> ps ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1333   = wceq 1335   F/wnf 1440   e. wcel 2128   A.wral 2435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714

This theorem is referenced by:  sumdc2  13333