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Theorem sbcralg 3235
 Description: Interchange class substitution and restricted quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
sbcralg
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   (,)   ()   ()   (,)

Proof of Theorem sbcralg
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3164 . 2
2 dfsbcq2 3164 . . 3
32ralbidv 2725 . 2
4 nfcv 2572 . . . 4
5 nfs1v 2182 . . . 4
64, 5nfral 2759 . . 3
7 sbequ12 1944 . . . 4
87ralbidv 2725 . . 3
96, 8sbie 2149 . 2
101, 3, 9vtoclbg 3012 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wceq 1652  wsb 1658   wcel 1725  wral 2705  wsbc 3161 This theorem is referenced by:  r19.12sn  3872  rspsbc2  28618  rspsbc2VD  28967  cdlemkid3N  31730  cdlemkid4  31731 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 2417 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 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-v 2958  df-sbc 3162
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