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Theorem zfcndreg 8943
Description: Axiom of Regularity ax-reg 7970, reproved from conditionless ZFC axioms. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)
Assertion
Ref Expression
zfcndreg  |-  ( E. y  y  e.  x  ->  E. y ( y  e.  x  /\  A. z ( z  e.  y  ->  -.  z  e.  x ) ) )
Distinct variable group:    x, y, z

Proof of Theorem zfcndreg
StepHypRef Expression
1 nfe1 1862 . 2  |-  F/ y E. y ( y  e.  x  /\  A. z ( z  e.  y  ->  -.  z  e.  x ) )
2 axregnd 8929 . 2  |-  ( y  e.  x  ->  E. y
( y  e.  x  /\  A. z ( z  e.  y  ->  -.  z  e.  x )
) )
31, 2exlimi 1938 1  |-  ( E. y  y  e.  x  ->  E. y ( y  e.  x  /\  A. z ( z  e.  y  ->  -.  z  e.  x ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367   A.wal 1401   E.wex 1631    e. wcel 1840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pr 4627  ax-reg 7970
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-v 3058  df-dif 3414  df-un 3416  df-nul 3736  df-sn 3970  df-pr 3972
This theorem is referenced by: (None)
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