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Theorem axc16nf 2265
 Description: If dtru 5248 is false, then there is only one element in the universe, so everything satisfies Ⅎ. (Contributed by Mario Carneiro, 7-Oct-2016.) Remove dependency on ax-11 2161. (Revised by Wolf Lammen, 9-Sep-2018.) (Proof shortened by BJ, 14-Jun-2019.) Remove dependency on ax-10 2145. (Revised by Wolf lammen, 12-Oct-2021.)
Assertion
Ref Expression
axc16nf (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem axc16nf
StepHypRef Expression
1 axc16g 2262 . . . 4 (∀𝑥 𝑥 = 𝑦 → (¬ 𝜑 → ∀𝑧 ¬ 𝜑))
2 eximal 1784 . . . 4 ((∃𝑧𝜑𝜑) ↔ (¬ 𝜑 → ∀𝑧 ¬ 𝜑))
31, 2sylibr 237 . . 3 (∀𝑥 𝑥 = 𝑦 → (∃𝑧𝜑𝜑))
4 axc16g 2262 . . 3 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
53, 4syld 47 . 2 (∀𝑥 𝑥 = 𝑦 → (∃𝑧𝜑 → ∀𝑧𝜑))
65nfd 1792 1 (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1536  ∃wex 1781  Ⅎwnf 1785 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  ax-12 2178 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786 This theorem is referenced by:  nfsbd  2564  nfsbOLD  2566  exists2  2748
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