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Theorem axc16nf 2191
 Description: If dtru 5128 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 2094. (Revised by Wolf Lammen, 9-Sep-2018.) (Proof shortened by BJ, 14-Jun-2019.) Remove dependency on ax-10 2080. (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 df-ex 1744 . . . 4 (∃𝑧𝜑 ↔ ¬ ∀𝑧 ¬ 𝜑)
2 axc16g 2188 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (¬ 𝜑 → ∀𝑧 ¬ 𝜑))
32con1d 142 . . . 4 (∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑧 ¬ 𝜑𝜑))
41, 3syl5bi 234 . . 3 (∀𝑥 𝑥 = 𝑦 → (∃𝑧𝜑𝜑))
5 axc16g 2188 . . 3 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
64, 5syld 47 . 2 (∀𝑥 𝑥 = 𝑦 → (∃𝑧𝜑 → ∀𝑧𝜑))
76nfd 1754 1 (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1506  ∃wex 1743  Ⅎwnf 1747 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-12 2107 This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1744  df-nf 1748 This theorem is referenced by:  nfsb  2490  nfsbd  2492  exists2  2703
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