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Theorem nd3 8301
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 7401 . . . 4  |-  -.  x  e.  x
2 elequ2 1715 . . . 4  |-  ( x  =  y  ->  (
x  e.  x  <->  x  e.  y ) )
31, 2mtbii 293 . . 3  |-  ( x  =  y  ->  -.  x  e.  y )
43sps 1755 . 2  |-  ( A. x  x  =  y  ->  -.  x  e.  y )
5 sp 1748 . 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 1540    = wceq 1642    e. wcel 1710
This theorem is referenced by:  nd4  8302  axrepnd  8306  axpowndlem3  8311  axinfnd  8318  axacndlem3  8321  axacnd  8324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295  ax-reg 7396
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-v 2866  df-dif 3231  df-un 3233  df-nul 3532  df-sn 3722  df-pr 3723
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