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Theorem nd3 8981
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 8041 . . . 4  |-  -.  x  e.  x
2 elequ2 1824 . . . 4  |-  ( x  =  y  ->  (
x  e.  x  <->  x  e.  y ) )
31, 2mtbii 302 . . 3  |-  ( x  =  y  ->  -.  x  e.  y )
43sps 1866 . 2  |-  ( A. x  x  =  y  ->  -.  x  e.  y )
5 sp 1860 . 2  |-  ( A. z  x  e.  y  ->  x  e.  y )
64, 5nsyl 121 1  |-  ( A. x  x  =  y  ->  -.  A. z  x  e.  y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-reg 8036
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-v 3111  df-dif 3474  df-un 3476  df-nul 3794  df-sn 4033  df-pr 4035
This theorem is referenced by:  nd4  8982  axrepnd  8986  axpowndlem3  8992  axpowndlem3OLD  8993  axinfnd  9001  axacndlem3  9004  axacnd  9007
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