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Theorem nd2 10354
Description: A lemma for proving conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2372. (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 9342 . . 3 ¬ 𝑧𝑧
2 stdpc4 2071 . . . 4 (∀𝑦 𝑧𝑦 → [𝑧 / 𝑦]𝑧𝑦)
31nfnth 1805 . . . . 5 𝑦 𝑧𝑧
4 elequ2 2121 . . . . 5 (𝑦 = 𝑧 → (𝑧𝑦𝑧𝑧))
53, 4sbie 2506 . . . 4 ([𝑧 / 𝑦]𝑧𝑦𝑧𝑧)
62, 5sylib 217 . . 3 (∀𝑦 𝑧𝑦𝑧𝑧)
71, 6mto 196 . 2 ¬ ∀𝑦 𝑧𝑦
8 axc11 2430 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑧𝑦 → ∀𝑦 𝑧𝑦))
97, 8mtoi 198 1 (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑧𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1537  [wsb 2067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-13 2372  ax-ext 2709  ax-sep 5221  ax-nul 5228  ax-pr 5350  ax-reg 9338
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-v 3431  df-dif 3889  df-un 3891  df-nul 4257  df-sn 4562  df-pr 4564
This theorem is referenced by:  axrepnd  10360  axpownd  10367  axinfndlem1  10371  axacndlem4  10376
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