MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  axc16nf Structured version   Visualization version   GIF version

Theorem axc16nf 2133
Description: If dtru 4817 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 2031. (Revised by Wolf Lammen, 9-Sep-2018.) (Proof shortened by BJ, 14-Jun-2019.) Remove dependency on ax-10 2016. (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 1702 . . . 4 (∃𝑧𝜑 ↔ ¬ ∀𝑧 ¬ 𝜑)
2 axc16g 2130 . . . . 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 2130 . . 3 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
64, 5syld 47 . 2 (∀𝑥 𝑥 = 𝑦 → (∃𝑧𝜑 → ∀𝑧𝜑))
76nfd 1713 1 (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1478  wex 1701  wnf 1705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-nf 1707
This theorem is referenced by:  nfsb  2439  nfsbd  2441
  Copyright terms: Public domain W3C validator