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Theorem nd1 10481
Description: A lemma for proving conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2371. (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 9490 . . 3 ¬ 𝑧𝑧
2 stdpc4 2071 . . . 4 (∀𝑦 𝑦𝑧 → [𝑧 / 𝑦]𝑦𝑧)
31nfnth 1804 . . . . 5 𝑦 𝑧𝑧
4 elequ1 2113 . . . . 5 (𝑦 = 𝑧 → (𝑦𝑧𝑧𝑧))
53, 4sbie 2505 . . . 4 ([𝑧 / 𝑦]𝑦𝑧𝑧𝑧)
62, 5sylib 217 . . 3 (∀𝑦 𝑦𝑧𝑧𝑧)
71, 6mto 196 . 2 ¬ ∀𝑦 𝑦𝑧
8 axc11 2429 . 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 1539  [wsb 2067
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-13 2371  ax-ext 2708  ax-sep 5254  ax-pr 5382  ax-reg 9486
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3063  df-rex 3072  df-v 3445  df-un 3913  df-sn 4585  df-pr 4587
This theorem is referenced by:  axrepnd  10488  axinfndlem1  10499  axinfnd  10500  axacndlem1  10501  axacndlem2  10502
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