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Theorem axc16nf 2175
Description: If dtru 4887 is false, then there is only one element in the universe, so everything satisfies . (Contributed by Mario Carneiro, 7-Oct-2016.) Remove dependency on ax-11 2074. (Revised by Wolf Lammen, 9-Sep-2018.) (Proof shortened by BJ, 14-Jun-2019.) Remove dependency on ax-10 2059. (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 1745 . . . 4 (∃𝑧𝜑 ↔ ¬ ∀𝑧 ¬ 𝜑)
2 axc16g 2172 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (¬ 𝜑 → ∀𝑧 ¬ 𝜑))
32con1d 139 . . . 4 (∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑧 ¬ 𝜑𝜑))
41, 3syl5bi 232 . . 3 (∀𝑥 𝑥 = 𝑦 → (∃𝑧𝜑𝜑))
5 axc16g 2172 . . 3 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
64, 5syld 47 . 2 (∀𝑥 𝑥 = 𝑦 → (∃𝑧𝜑 → ∀𝑧𝜑))
76nfd 1756 1 (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1521  wex 1744  wnf 1748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-nf 1750
This theorem is referenced by:  nfsb  2468  nfsbd  2470
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