Theorem rspct 3203

Description: A closed version of rspc 3204. (Contributed by Andrew Salmon, 6-Jun-2011.)

Hypothesis

Ref	Expression
rspct.1

Assertion

Ref	Expression
rspct

Distinct variable groups:   ,   ,

Proof of Theorem rspct

Step	Hyp	Ref	Expression
1		df-ral 2812	. . . 4
2		eleq1 2529	. . . . . . . . . 10
3	2	adantr 465	. . . . . . . . 9
4		simpr 461	. . . . . . . . 9
5	3, 4	imbi12d 320	. . . . . . . 8
6	5	ex 434	. . . . . . 7
7	6	a2i 13	. . . . . 6
8	7	alimi 1633	. . . . 5
9		nfv 1707	. . . . . . 7
10		rspct.1	. . . . . . 7
11	9, 10	nfim 1920	. . . . . 6
12		nfcv 2619	. . . . . 6
13	11, 12	spcgft 3186	. . . . 5
14	8, 13	syl 16	. . . 4
15	1, 14	syl7bi 230	. . 3
16	15	com34 83	. 2
17	16	pm2.43d 48	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  =wceq 1395  F/wnf 1616  e.wcel 1818  A.wral 2807

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-v 3111