MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nd2 Structured version   Visualization version   GIF version

Theorem nd2 10008
Description: A lemma for proving conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2392. (Contributed by NM, 1-Jan-2002.) (New usage is discouraged.)
Assertion
Ref Expression
nd2 (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑧𝑦)

Proof of Theorem nd2
StepHypRef Expression
1 elirrv 9057 . . 3 ¬ 𝑧𝑧
2 stdpc4 2074 . . . 4 (∀𝑦 𝑧𝑦 → [𝑧 / 𝑦]𝑧𝑦)
31nfnth 1804 . . . . 5 𝑦 𝑧𝑧
4 elequ2 2130 . . . . 5 (𝑦 = 𝑧 → (𝑧𝑦𝑧𝑧))
53, 4sbie 2546 . . . 4 ([𝑧 / 𝑦]𝑧𝑦𝑧𝑧)
62, 5sylib 221 . . 3 (∀𝑦 𝑧𝑦𝑧𝑧)
71, 6mto 200 . 2 ¬ ∀𝑦 𝑧𝑦
8 axc11 2454 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑧𝑦 → ∀𝑦 𝑧𝑦))
97, 8mtoi 202 1 (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑧𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1536  [wsb 2070
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-13 2392  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317  ax-reg 9053
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 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-dif 3922  df-un 3924  df-nul 4277  df-sn 4551  df-pr 4553
This theorem is referenced by:  axrepnd  10014  axpownd  10021  axinfndlem1  10025  axacndlem4  10030
  Copyright terms: Public domain W3C validator