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Theorem nd3 8469
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 7568 . . . 4  |-  -.  x  e.  x
2 elequ2 1731 . . . 4  |-  ( x  =  y  ->  (
x  e.  x  <->  x  e.  y ) )
31, 2mtbii 295 . . 3  |-  ( x  =  y  ->  -.  x  e.  y )
43sps 1771 . 2  |-  ( A. x  x  =  y  ->  -.  x  e.  y )
5 sp 1764 . 2  |-  ( A. z  x  e.  y  ->  x  e.  y )
64, 5nsyl 116 1  |-  ( A. x  x  =  y  ->  -.  A. z  x  e.  y )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1550
This theorem is referenced by:  nd4  8470  axrepnd  8474  axpowndlem3  8479  axinfnd  8486  axacndlem3  8489  axacnd  8492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406  ax-reg 7563
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-v 2960  df-dif 3325  df-un 3327  df-nul 3631  df-sn 3822  df-pr 3823
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