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Theorem nd3 10000
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 9048 . . . 4 ¬ 𝑥𝑥
2 elequ2 2129 . . . 4 (𝑥 = 𝑦 → (𝑥𝑥𝑥𝑦))
31, 2mtbii 329 . . 3 (𝑥 = 𝑦 → ¬ 𝑥𝑦)
43sps 2185 . 2 (∀𝑥 𝑥 = 𝑦 → ¬ 𝑥𝑦)
5 sp 2183 . 2 (∀𝑧 𝑥𝑦𝑥𝑦)
64, 5nsyl 142 1 (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑧 𝑥𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307  ax-reg 9044
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-dif 3911  df-un 3913  df-nul 4266  df-sn 4540  df-pr 4542
This theorem is referenced by:  nd4  10001  axrepnd  10005  axpowndlem3  10010  axinfnd  10017  axacndlem3  10020  axacnd  10023
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