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Theorem nd3 8211
Description: A lemma for proving conditionless ZFC axioms. (Contributed by NM, 2-Jan-2002.)
Assertion
Ref Expression
nd3  |-  ( A. x  x  =  y  ->  -.  A. z  x  e.  y )

Proof of Theorem nd3
StepHypRef Expression
1 elirrv 7311 . . . 4  |-  -.  x  e.  x
2 elequ2 1689 . . . 4  |-  ( x  =  y  ->  (
x  e.  x  <->  x  e.  y ) )
31, 2mtbii 293 . . 3  |-  ( x  =  y  ->  -.  x  e.  y )
43sps 1739 . 2  |-  ( A. x  x  =  y  ->  -.  x  e.  y )
5 sp 1716 . 2  |-  ( A. z  x  e.  y  ->  x  e.  y )
64, 5nsyl 113 1  |-  ( A. x  x  =  y  ->  -.  A. z  x  e.  y )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1527    = wceq 1623    e. wcel 1684
This theorem is referenced by:  nd4  8212  axrepnd  8216  axpowndlem3  8221  axinfnd  8228  axacndlem3  8231  axacnd  8234
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-reg 7306
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-v 2790  df-dif 3155  df-un 3157  df-nul 3456  df-sn 3646  df-pr 3647
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