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Theorem nd2 9594
Description: A lemma for proving conditionless ZFC axioms. (Contributed by NM, 1-Jan-2002.)
Assertion
Ref Expression
nd2 (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑧𝑦)

Proof of Theorem nd2
StepHypRef Expression
1 elirrv 8658 . . 3 ¬ 𝑧𝑧
2 stdpc4 2482 . . . 4 (∀𝑦 𝑧𝑦 → [𝑧 / 𝑦]𝑧𝑦)
31nfnth 1869 . . . . 5 𝑦 𝑧𝑧
4 elequ2 2145 . . . . 5 (𝑦 = 𝑧 → (𝑧𝑦𝑧𝑧))
53, 4sbie 2537 . . . 4 ([𝑧 / 𝑦]𝑧𝑦𝑧𝑧)
62, 5sylib 208 . . 3 (∀𝑦 𝑧𝑦𝑧𝑧)
71, 6mto 188 . 2 ¬ ∀𝑦 𝑧𝑦
8 axc11 2448 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑧𝑦 → ∀𝑦 𝑧𝑦))
97, 8mtoi 190 1 (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑧𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1622  [wsb 2038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047  ax-reg 8654
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-v 3334  df-dif 3710  df-un 3712  df-nul 4051  df-sn 4314  df-pr 4316
This theorem is referenced by:  axrepnd  9600  axpownd  9607  axinfndlem1  9611  axacndlem4  9616
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