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Theorem nd1 9447
Description: A lemma for proving conditionless ZFC axioms. (Contributed by NM, 1-Jan-2002.)
Assertion
Ref Expression
nd1 (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑦𝑧)

Proof of Theorem nd1
StepHypRef Expression
1 elirrv 8542 . . 3 ¬ 𝑧𝑧
2 stdpc4 2381 . . . 4 (∀𝑦 𝑦𝑧 → [𝑧 / 𝑦]𝑦𝑧)
31nfnth 1768 . . . . 5 𝑦 𝑧𝑧
4 elequ1 2037 . . . . 5 (𝑦 = 𝑧 → (𝑦𝑧𝑧𝑧))
53, 4sbie 2436 . . . 4 ([𝑧 / 𝑦]𝑦𝑧𝑧𝑧)
62, 5sylib 208 . . 3 (∀𝑦 𝑦𝑧𝑧𝑧)
71, 6mto 188 . 2 ¬ ∀𝑦 𝑦𝑧
8 axc11 2347 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑦𝑧 → ∀𝑦 𝑦𝑧))
97, 8mtoi 190 1 (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑦𝑧)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1521  [wsb 1937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-reg 8538
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-dif 3610  df-un 3612  df-nul 3949  df-sn 4211  df-pr 4213
This theorem is referenced by:  axrepnd  9454  axinfndlem1  9465  axinfnd  9466  axacndlem1  9467  axacndlem2  9468
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