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Theorem sbcrexg 3121
 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 3049 . 2
2 dfsbcq2 3049 . . 3
32rexbidv 2635 . 2
4 nfcv 2489 . . . 4
5 nfs1v 2106 . . . 4
64, 5nfrex 2669 . . 3
7 sbequ12 1919 . . . 4
87rexbidv 2635 . . 3
96, 8sbie 2038 . 2
101, 3, 9vtoclbg 2915 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176   wceq 1642  wsb 1648   wcel 1710  wrex 2615  wsbc 3046 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047 This theorem is referenced by: (None)
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