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Theorem nd3 8424
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 7525 . . . 4  |-  -.  x  e.  x
2 elequ2 1726 . . . 4  |-  ( x  =  y  ->  (
x  e.  x  <->  x  e.  y ) )
31, 2mtbii 294 . . 3  |-  ( x  =  y  ->  -.  x  e.  y )
43sps 1766 . 2  |-  ( A. x  x  =  y  ->  -.  x  e.  y )
5 sp 1759 . 2  |-  ( A. z  x  e.  y  ->  x  e.  y )
64, 5nsyl 115 1  |-  ( A. x  x  =  y  ->  -.  A. z  x  e.  y )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1546
This theorem is referenced by:  nd4  8425  axrepnd  8429  axpowndlem3  8434  axinfnd  8441  axacndlem3  8444  axacnd  8447
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367  ax-reg 7520
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-v 2922  df-dif 3287  df-un 3289  df-nul 3593  df-sn 3784  df-pr 3785
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