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Theorem nd3 9290
 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 8387 . . . 4 ¬ 𝑥𝑥
2 elequ2 1991 . . . 4 (𝑥 = 𝑦 → (𝑥𝑥𝑥𝑦))
31, 2mtbii 315 . . 3 (𝑥 = 𝑦 → ¬ 𝑥𝑦)
43sps 2043 . 2 (∀𝑥 𝑥 = 𝑦 → ¬ 𝑥𝑦)
5 sp 2041 . 2 (∀𝑧 𝑥𝑦𝑥𝑦)
64, 5nsyl 134 1 (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑧 𝑥𝑦)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1473 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-reg 8380 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-un 3545  df-nul 3875  df-sn 4126  df-pr 4128 This theorem is referenced by:  nd4  9291  axrepnd  9295  axpowndlem3  9300  axinfnd  9307  axacndlem3  9310  axacnd  9313
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