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Theorem nd1 10621
Description: A lemma for proving conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2366. (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 9632 . . 3 ¬ 𝑧𝑧
2 stdpc4 2064 . . . 4 (∀𝑦 𝑦𝑧 → [𝑧 / 𝑦]𝑦𝑧)
31nfnth 1797 . . . . 5 𝑦 𝑧𝑧
4 elequ1 2106 . . . . 5 (𝑦 = 𝑧 → (𝑦𝑧𝑧𝑧))
53, 4sbie 2496 . . . 4 ([𝑧 / 𝑦]𝑦𝑧𝑧𝑧)
62, 5sylib 217 . . 3 (∀𝑦 𝑦𝑧𝑧𝑧)
71, 6mto 196 . 2 ¬ ∀𝑦 𝑦𝑧
8 axc11 2424 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑦𝑧 → ∀𝑦 𝑦𝑧))
97, 8mtoi 198 1 (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑦𝑧)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1532  [wsb 2060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-13 2366  ax-ext 2697  ax-sep 5296  ax-pr 5425  ax-reg 9628
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-v 3464  df-un 3951  df-sn 4624  df-pr 4626
This theorem is referenced by:  axrepnd  10628  axinfndlem1  10639  axinfnd  10640  axacndlem1  10641  axacndlem2  10642
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