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Theorem bj-axc14nf 33172
Description: Proof of a version of axc14 2519 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 bj-nfeel2 33171 . 2 (¬ ∀𝑧 𝑧 = 𝑥 → Ⅎ𝑧 𝑥𝑡)
2 elequ2 2159 . 2 (𝑡 = 𝑦 → (𝑥𝑡𝑥𝑦))
31, 2bj-dvelimdv1 33169 1 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧 𝑥𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1629  wnf 1856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858
This theorem is referenced by:  bj-axc14  33173
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