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Theorem zfregcl 7554
 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 2496 . . . 4
32exbidv 1636 . . 3
4 eleq2 2496 . . . . . 6
54notbid 286 . . . . 5
65ralbidv 2717 . . . 4
76rexeqbi1dv 2905 . . 3
83, 7imbi12d 312 . 2
9 nfre1 2754 . . 3
10 axreg2 7553 . . . 4
11 df-ral 2702 . . . . . 6
1211rexbii 2722 . . . . 5
13 df-rex 2703 . . . . 5
1412, 13bitr2i 242 . . . 4
1510, 14sylib 189 . . 3
169, 15exlimi 1821 . 2
171, 8, 16vtocl 2998 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 359  wal 1549  wex 1550   wceq 1652   wcel 1725  wral 2697  wrex 2698  cvv 2948 This theorem is referenced by:  zfreg  7555  zfreg2  7556  elirrv  7557 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  ax-reg 7552 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
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