Theorem rspct 3302

Description: A closed version of rspc 3303. (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 2917	. . . 4 |- ( A. x e. B ph <-> A. x ( x e. B -> ph ) )
2		eleq1 2689	. . . . . . . . . 10 |- ( x = A -> ( x e. B <-> A e. B ) )
3	2	adantr 481	. . . . . . . . 9 |- ( ( x = A /\ ( ph <-> ps ) ) -> ( x e. B <-> A e. B ) )
4		simpr 477	. . . . . . . . 9 |- ( ( x = A /\ ( ph <-> ps ) ) -> ( ph <-> ps ) )
5	3, 4	imbi12d 334	. . . . . . . 8 |- ( ( x = A /\ ( ph <-> ps ) ) -> ( ( x e. B -> ph ) <-> ( A e. B -> ps ) ) )
6	5	ex 450	. . . . . . 7 |- ( x = A -> ( ( ph <-> ps ) -> ( ( x e. B -> ph ) <-> ( A e. B -> ps ) ) ) )
7	6	a2i 14	. . . . . 6 |- ( ( x = A -> ( ph <-> ps ) ) -> ( x = A -> ( ( x e. B -> ph ) <-> ( A e. B -> ps ) ) ) )
8	7	alimi 1739	. . . . 5 |- ( A. x ( x = A -> ( ph <-> ps ) ) -> A. x ( x = A -> ( ( x e. B -> ph ) <-> ( A e. B -> ps ) ) ) )
9		nfv 1843	. . . . . . 7 |- F/ x A e. B
10		rspct.1	. . . . . . 7 |- F/ x ps
11	9, 10	nfim 1825	. . . . . 6 |- F/ x ( A e. B -> ps )
12		nfcv 2764	. . . . . 6 |- F/_ x A
13	11, 12	spcgft 3285	. . . . 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 17	. . . 4 |- ( A. x ( x = A -> ( ph <-> ps ) ) -> ( A e. B -> ( A. x ( x e. B -> ph ) -> ( A e. B -> ps ) ) ) )
15	1, 14	syl7bi 245	. . 3 |- ( A. x ( x = A -> ( ph <-> ps ) ) -> ( A e. B -> ( A. x e. B ph -> ( A e. B -> ps ) ) ) )
16	15	com34 91	. 2 |- ( A. x ( x = A -> ( ph <-> ps ) ) -> ( A e. B -> ( A e. B -> ( A. x e. B ph -> ps ) ) ) )
17	16	pm2.43d 53	1 |- ( A. x ( x = A -> ( ph <-> ps ) ) -> ( A e. B -> ( A. x e. B ph -> ps ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   F/wnf 1708   e. wcel 1990   A.wral 2912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202

This theorem is referenced by:  rspcdf  42424