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Theorem nd3 10580
Description: A lemma for proving conditionless ZFC axioms. (Contributed by NM, 2-Jan-2002.)
Assertion
Ref Expression
nd3 (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑧 𝑥𝑦)

Proof of Theorem nd3
StepHypRef Expression
1 elirrv 9587 . . . 4 ¬ 𝑥𝑥
2 elequ2 2113 . . . 4 (𝑥 = 𝑦 → (𝑥𝑥𝑥𝑦))
31, 2mtbii 326 . . 3 (𝑥 = 𝑦 → ¬ 𝑥𝑦)
43sps 2170 . 2 (∀𝑥 𝑥 = 𝑦 → ¬ 𝑥𝑦)
5 sp 2168 . 2 (∀𝑧 𝑥𝑦𝑥𝑦)
64, 5nsyl 140 1 (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑧 𝑥𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-pr 5417  ax-reg 9583
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-v 3468  df-un 3945  df-sn 4621  df-pr 4623
This theorem is referenced by:  nd4  10581  axrepnd  10585  axpowndlem3  10590  axinfnd  10597  axacndlem3  10600  axacnd  10603
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