Theorem sbcrexgOLD 3413

Description: Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) Obsolete as of 18-Aug-2018. Use sbcrex 3412 instead. (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
sbcrexgOLD

Distinct variable groups:   ,   ,   ,

Proof of Theorem sbcrexgOLD

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 3330	. 2
2		dfsbcq2 3330	. . 3
3	2	rexbidv 2968	. 2
4		nfcv 2619	. . . 4
5		nfs1v 2181	. . . 4
6	4, 5	nfrex 2920	. . 3
7		sbequ12 1992	. . . 4
8	7	rexbidv 2968	. . 3
9	6, 8	sbie 2149	. 2
10	1, 3, 9	vtoclbg 3168	1

Syntax hints:  ->wi 4  <->wb 184  =wceq 1395  [wsb 1739  e.wcel 1818  E.wrex 2808  [.wsbc 3327

This theorem is referenced by:  2sbcrexOLD  30719  sbc2rexgOLD  30721

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-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