Theorem wl-hbnaev 33305

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 2019. (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.

Step	Hyp	Ref	Expression
1		aev 1983	. . 3 ⊢ (∀𝑢 𝑢 = 𝑡 → ∀𝑥 𝑥 = 𝑦)
2	1	con3i 150	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑢 𝑢 = 𝑡)
3		ax-5 1839	. 2 ⊢ (¬ ∀𝑢 𝑢 = 𝑡 → ∀𝑧 ¬ ∀𝑢 𝑢 = 𝑡)
4		aev 1983	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑢 𝑢 = 𝑡)
5	4	con3i 150	. . 3 ⊢ (¬ ∀𝑢 𝑢 = 𝑡 → ¬ ∀𝑥 𝑥 = 𝑦)
6	5	alimi 1739	. 2 ⊢ (∀𝑧 ¬ ∀𝑢 𝑢 = 𝑡 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)
7	2, 3, 6	3syl 18	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by: (None)