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Theorem bj-axc14nf 33749
 Description: Proof of a version of axc14 2443 using the "nonfree" 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 bj-nfeel2 33748 . 2 (¬ ∀𝑧 𝑧 = 𝑥 → Ⅎ𝑧 𝑥𝑡)
2 elequ2 2096 . 2 (𝑡 = 𝑦 → (𝑥𝑡𝑥𝑦))
31, 2bj-dvelimdv1 33746 1 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧 𝑥𝑦))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1520  Ⅎwnf 1765 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766 This theorem is referenced by:  bj-axc14  33750
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