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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.
StepHypRef Expression
1 dfsbcq2 3330 . 2
2 dfsbcq2 3330 . . 3
32rexbidv 2968 . 2
4 nfcv 2619 . . . 4
5 nfs1v 2181 . . . 4
64, 5nfrex 2920 . . 3
7 sbequ12 1992 . . . 4
87rexbidv 2968 . . 3
96, 8sbie 2149 . 2
101, 3, 9vtoclbg 3168 1
Colors of variables: wff setvar class
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
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