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Theorem nd2 10626
Description: A lemma for proving conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2375. (Contributed by NM, 1-Jan-2002.) (New usage is discouraged.)
Assertion
Ref Expression
nd2 (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑧𝑦)

Proof of Theorem nd2
StepHypRef Expression
1 elirrv 9634 . . 3 ¬ 𝑧𝑧
2 stdpc4 2066 . . . 4 (∀𝑦 𝑧𝑦 → [𝑧 / 𝑦]𝑧𝑦)
31nfnth 1799 . . . . 5 𝑦 𝑧𝑧
4 elequ2 2121 . . . . 5 (𝑦 = 𝑧 → (𝑧𝑦𝑧𝑧))
53, 4sbie 2505 . . . 4 ([𝑧 / 𝑦]𝑧𝑦𝑧𝑧)
62, 5sylib 218 . . 3 (∀𝑦 𝑧𝑦𝑧𝑧)
71, 6mto 197 . 2 ¬ ∀𝑦 𝑧𝑦
8 axc11 2433 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑧𝑦 → ∀𝑦 𝑧𝑦))
97, 8mtoi 199 1 (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑧𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1535  [wsb 2062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-13 2375  ax-ext 2706  ax-sep 5302  ax-pr 5438  ax-reg 9630
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-v 3480  df-un 3968  df-sn 4632  df-pr 4634
This theorem is referenced by:  axrepnd  10632  axpownd  10639  axinfndlem1  10643  axacndlem4  10648
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