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Theorem sbcrexg 3228
 Description: Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
sbcrexg
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   (,)   ()   ()   (,)

Proof of Theorem sbcrexg
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3156 . 2
2 dfsbcq2 3156 . . 3
32rexbidv 2718 . 2
4 nfcv 2571 . . . 4
5 nfs1v 2181 . . . 4
64, 5nfrex 2753 . . 3
7 sbequ12 1944 . . . 4
87rexbidv 2718 . . 3
96, 8sbie 2122 . 2
101, 3, 9vtoclbg 3004 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wceq 1652  wsb 1658   wcel 1725  wrex 2698  wsbc 3153 This theorem is referenced by:  ac6sfi  7343  rexfiuz  12143  2sbcrex  26834  sbc2rexg  26835  4rexfrabdioph  26849 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-v 2950  df-sbc 3154
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