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Theorem nd1 10002
Description: A lemma for proving conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2382. (Contributed by NM, 1-Jan-2002.) (New usage is discouraged.)
Assertion
Ref Expression
nd1 (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑦𝑧)

Proof of Theorem nd1
StepHypRef Expression
1 elirrv 9048 . . 3 ¬ 𝑧𝑧
2 stdpc4 2073 . . . 4 (∀𝑦 𝑦𝑧 → [𝑧 / 𝑦]𝑦𝑧)
31nfnth 1804 . . . . 5 𝑦 𝑧𝑧
4 elequ1 2119 . . . . 5 (𝑦 = 𝑧 → (𝑦𝑧𝑧𝑧))
53, 4sbie 2524 . . . 4 ([𝑧 / 𝑦]𝑦𝑧𝑧𝑧)
62, 5sylib 221 . . 3 (∀𝑦 𝑦𝑧𝑧𝑧)
71, 6mto 200 . 2 ¬ ∀𝑦 𝑦𝑧
8 axc11 2444 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑦𝑧 → ∀𝑦 𝑦𝑧))
97, 8mtoi 202 1 (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑦𝑧)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1536  [wsb 2069
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 2114  ax-9 2122  ax-10 2143  ax-12 2176  ax-13 2382  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298  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 2780  df-cleq 2794  df-clel 2873  df-ral 3114  df-rex 3115  df-v 3446  df-dif 3887  df-un 3889  df-nul 4247  df-sn 4529  df-pr 4531
This theorem is referenced by:  axrepnd  10009  axinfndlem1  10020  axinfnd  10021  axacndlem1  10022  axacndlem2  10023
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