Theorem rspct 2834

Description: A closed version of rspc 2835. (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 2460	. . . 4 |- ( A. x e. B ph <-> A. x ( x e. B -> ph ) )
2		eleq1 2240	. . . . . . . . . 10 |- ( x = A -> ( x e. B <-> A e. B ) )
3	2	adantr 276	. . . . . . . . 9 |- ( ( x = A /\ ( ph <-> ps ) ) -> ( x e. B <-> A e. B ) )
4		simpr 110	. . . . . . . . 9 |- ( ( x = A /\ ( ph <-> ps ) ) -> ( ph <-> ps ) )
5	3, 4	imbi12d 234	. . . . . . . 8 |- ( ( x = A /\ ( ph <-> ps ) ) -> ( ( x e. B -> ph ) <-> ( A e. B -> ps ) ) )
6	5	ex 115	. . . . . . 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 1455	. . . . 5 |- ( A. x ( x = A -> ( ph <-> ps ) ) -> A. x ( x = A -> ( ( x e. B -> ph ) <-> ( A e. B -> ps ) ) ) )
9		nfv 1528	. . . . . . 7 |- F/ x A e. B
10		rspct.1	. . . . . . 7 |- F/ x ps
11	9, 10	nfim 1572	. . . . . 6 |- F/ x ( A e. B -> ps )
12		nfcv 2319	. . . . . 6 |- F/_ x A
13	11, 12	spcgft 2814	. . . . 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 165	. . 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 104   <-> wb 105   A.wal 1351   = wceq 1353   F/wnf 1460   e. wcel 2148   A.wral 2455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739

This theorem is referenced by:  sumdc2  14402