Theorem naev2 2064

Description: Generalization of hbnaev 2065. (Contributed by Wolf Lammen, 9-Apr-2021.)

Assertion

Ref	Expression
naev2	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑡 𝑡 = 𝑢)

Distinct variable group:   𝑢,𝑡

Proof of Theorem naev2

Dummy variables 𝑤 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		naev 2063	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑣 𝑣 = 𝑤)
2		ax-5 1911	. 2 ⊢ (¬ ∀𝑣 𝑣 = 𝑤 → ∀𝑧 ¬ ∀𝑣 𝑣 = 𝑤)
3		naev 2063	. . 3 ⊢ (¬ ∀𝑣 𝑣 = 𝑤 → ¬ ∀𝑡 𝑡 = 𝑢)
4	3	alimi 1812	. 2 ⊢ (∀𝑧 ¬ ∀𝑣 𝑣 = 𝑤 → ∀𝑧 ¬ ∀𝑡 𝑡 = 𝑢)
5	1, 2, 4	3syl 18	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑡 𝑡 = 𝑢)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781

This theorem is referenced by:  hbnaev  2065