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Theorem zfregcl 8109
 Description: The Axiom of Regularity with class variables. (Contributed by NM, 5-Aug-1994.)
Hypothesis
Ref Expression
zfregcl.1
Assertion
Ref Expression
zfregcl
Distinct variable group:   ,,

Proof of Theorem zfregcl
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 zfregcl.1 . 2
2 eleq2 2518 . . . 4
32exbidv 1768 . . 3
4 eleq2 2518 . . . . . 6
54notbid 296 . . . . 5
65ralbidv 2827 . . . 4
76rexeqbi1dv 2996 . . 3
83, 7imbi12d 322 . 2
9 nfre1 2848 . . 3
10 axreg2 8108 . . . 4
11 df-ral 2742 . . . . . 6
1211rexbii 2889 . . . . 5
13 df-rex 2743 . . . . 5
1412, 13bitr2i 254 . . . 4
1510, 14sylib 200 . . 3
169, 15exlimi 1995 . 2
171, 8, 16vtocl 3100 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wa 371  wal 1442   wceq 1444  wex 1663   wcel 1887  wral 2737  wrex 2738  cvv 3045 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-ext 2431  ax-reg 8107 This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ral 2742  df-rex 2743  df-v 3047 This theorem is referenced by:  zfreg  8110  zfreg2  8111  elirrv  8112
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