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Theorem nd2 10629
Description: A lemma for proving conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2376. (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 9637 . . 3 ¬ 𝑧𝑧
2 stdpc4 2067 . . . 4 (∀𝑦 𝑧𝑦 → [𝑧 / 𝑦]𝑧𝑦)
31nfnth 1801 . . . . 5 𝑦 𝑧𝑧
4 elequ2 2122 . . . . 5 (𝑦 = 𝑧 → (𝑧𝑦𝑧𝑧))
53, 4sbie 2506 . . . 4 ([𝑧 / 𝑦]𝑧𝑦𝑧𝑧)
62, 5sylib 218 . . 3 (∀𝑦 𝑧𝑦𝑧𝑧)
71, 6mto 197 . 2 ¬ ∀𝑦 𝑧𝑦
8 axc11 2434 . 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 1537  [wsb 2063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-13 2376  ax-ext 2707  ax-sep 5295  ax-pr 5431  ax-reg 9633
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-v 3481  df-un 3955  df-sn 4626  df-pr 4628
This theorem is referenced by:  axrepnd  10635  axpownd  10642  axinfndlem1  10646  axacndlem4  10651
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