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Theorem naev2 2066
 Description: Generalization of hbnaev 2067. (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.
StepHypRef Expression
1 naev 2065 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑣 𝑣 = 𝑤)
2 ax-5 1911 . 2 (¬ ∀𝑣 𝑣 = 𝑤 → ∀𝑧 ¬ ∀𝑣 𝑣 = 𝑤)
3 naev 2065 . . 3 (¬ ∀𝑣 𝑣 = 𝑤 → ¬ ∀𝑡 𝑡 = 𝑢)
43alimi 1813 . 2 (∀𝑧 ¬ ∀𝑣 𝑣 = 𝑤 → ∀𝑧 ¬ ∀𝑡 𝑡 = 𝑢)
51, 2, 43syl 18 1 (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑡 𝑡 = 𝑢)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1536 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782 This theorem is referenced by:  hbnaev  2067
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