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Theorem bj-axc14nf 34685
Description: Proof of a version of axc14 2464 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 34684 . 2 (¬ ∀𝑧 𝑧 = 𝑥 → Ⅎ𝑧 𝑥𝑡)
2 elequ2 2129 . 2 (𝑡 = 𝑦 → (𝑥𝑡𝑥𝑦))
31, 2bj-dvelimdv1 34682 1 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧 𝑥𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1540  wnf 1790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-13 2373
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-nf 1791
This theorem is referenced by:  bj-axc14  34686
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