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Theorem nd3 9746
 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 8790 . . . 4 ¬ 𝑥𝑥
2 elequ2 2120 . . . 4 (𝑥 = 𝑦 → (𝑥𝑥𝑥𝑦))
31, 2mtbii 318 . . 3 (𝑥 = 𝑦 → ¬ 𝑥𝑦)
43sps 2168 . 2 (∀𝑥 𝑥 = 𝑦 → ¬ 𝑥𝑦)
5 sp 2166 . 2 (∀𝑧 𝑥𝑦𝑥𝑦)
64, 5nsyl 138 1 (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑧 𝑥𝑦)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1599 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138  ax-reg 8786 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-v 3399  df-dif 3794  df-un 3796  df-nul 4141  df-sn 4398  df-pr 4400 This theorem is referenced by:  nd4  9747  axrepnd  9751  axpowndlem3  9756  axinfnd  9763  axacndlem3  9766  axacnd  9769
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