Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-axc14nf Structured version   Visualization version   GIF version

Theorem bj-axc14nf 32963
Description: Proof of a version of axc14 2400 using the "non-free" idiom. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-axc14nf (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧 𝑥𝑦))

Proof of Theorem bj-axc14nf
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 nfnae 2351 . 2 𝑧 ¬ ∀𝑧 𝑧 = 𝑥
2 bj-nfeel2 32962 . 2 (¬ ∀𝑧 𝑧 = 𝑥 → Ⅎ𝑧 𝑥𝑡)
3 elequ2 2044 . 2 (𝑡 = 𝑦 → (𝑥𝑡𝑥𝑦))
41, 2, 3bj-dvelimdv1 32960 1 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧 𝑥𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1521  wnf 1748
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
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750
This theorem is referenced by:  bj-axc14  32964
  Copyright terms: Public domain W3C validator