Theorem sbcrextOLD 3409

Description: Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) Obsolete as of 18-Aug-2018. (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
sbcrextOLD

Distinct variable groups:   ,   ,

Proof of Theorem sbcrextOLD

Step	Hyp	Ref	Expression
1		elex 3118	. 2
2		sbcng 3368	. . . . 5
3	2	adantr 465	. . . 4
4		sbcralt 3408	. . . . . 6
5		nfnfc1 2622	. . . . . . . . 9
6		id 22	. . . . . . . . . 10
7		nfcvd 2620	. . . . . . . . . 10
8	6, 7	nfeld 2627	. . . . . . . . 9
9	5, 8	nfan1 1927	. . . . . . . 8
10		sbcng 3368	. . . . . . . . 9
11	10	adantl 466	. . . . . . . 8
12	9, 11	ralbid 2891	. . . . . . 7
13	12	ancoms 453	. . . . . 6
14	4, 13	bitrd 253	. . . . 5
15	14	notbid 294	. . . 4
16	3, 15	bitrd 253	. . 3
17		dfrex2 2908	. . . 4
18	17	sbcbii 3387	. . 3
19		dfrex2 2908	. . 3
20	16, 18, 19	3bitr4g 288	. 2
21	1, 20	sylan 471	1

Syntax hints:  -.wn 3  ->wi 4  <->wb 184  /\wa 369  e.wcel 1818  F/_wnfc 2605  A.wral 2807  E.wrex 2808  cvv 3109  [.wsbc 3327

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-or 370  df-an 371  df-3an 975  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-rex 2813  df-v 3111  df-sbc 3328