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Theorem nd3 8965
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 8065 . . . 4  |-  -.  x  e.  x
2 elequ2 1877 . . . 4  |-  ( x  =  y  ->  (
x  e.  x  <->  x  e.  y ) )
31, 2mtbii 303 . . 3  |-  ( x  =  y  ->  -.  x  e.  y )
43sps 1920 . 2  |-  ( A. x  x  =  y  ->  -.  x  e.  y )
5 sp 1914 . 2  |-  ( A. z  x  e.  y  ->  x  e.  y )
64, 5nsyl 124 1  |-  ( A. x  x  =  y  ->  -.  A. z  x  e.  y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pr 4603  ax-reg 8060
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-v 3024  df-dif 3382  df-un 3384  df-nul 3705  df-sn 3942  df-pr 3944
This theorem is referenced by:  nd4  8966  axrepnd  8970  axpowndlem3  8975  axinfnd  8982  axacndlem3  8985  axacnd  8988
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