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Theorem wl-hbnaev 33435
Description: Any variable is free in ¬ ∀𝑥𝑥 = 𝑦, if 𝑥 and 𝑦 are distinct. The latter condition can actually be lifted, but this version is easier to prove. The proof does not use ax-10 2059. (Contributed by Wolf Lammen, 9-Apr-2021.)
Assertion
Ref Expression
wl-hbnaev (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)
Distinct variable group:   𝑥,𝑦

Proof of Theorem wl-hbnaev
Dummy variables 𝑢 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 aev 2025 . . 3 (∀𝑢 𝑢 = 𝑡 → ∀𝑥 𝑥 = 𝑦)
21con3i 150 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑢 𝑢 = 𝑡)
3 ax-5 1879 . 2 (¬ ∀𝑢 𝑢 = 𝑡 → ∀𝑧 ¬ ∀𝑢 𝑢 = 𝑡)
4 aev 2025 . . . 4 (∀𝑥 𝑥 = 𝑦 → ∀𝑢 𝑢 = 𝑡)
54con3i 150 . . 3 (¬ ∀𝑢 𝑢 = 𝑡 → ¬ ∀𝑥 𝑥 = 𝑦)
65alimi 1779 . 2 (∀𝑧 ¬ ∀𝑢 𝑢 = 𝑡 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)
72, 3, 63syl 18 1 (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1521
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
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745
This theorem is referenced by: (None)
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