Theorem rspct 2777

Description: A closed version of rspc 2778. (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 2419	. . . 4 |- ( A. x e. B ph <-> A. x ( x e. B -> ph ) )
2		eleq1 2200	. . . . . . . . . 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 1431	. . . . 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 1551	. . . . . 6 |- F/ x ( A e. B -> ps )
12		nfcv 2279	. . . . . 6 |- F/_ x A
13	11, 12	spcgft 2758	. . . . 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 1329   = wceq 1331   F/wnf 1436   e. wcel 1480   A.wral 2414

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-v 2683

This theorem is referenced by:  sumdc2  12995