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

Theorem axbnd 2776
 Description: Axiom of Bundling (intuitionistic logic axiom ax-bnd). In classical logic, this and axi12 2775 are fairly straightforward consequences of axc9 2389. But in intuitionistic logic, it is not easy to add the extra ∀𝑥 to axi12 2775 and so we treat the two as separate axioms. (Contributed by Jim Kingdon, 22-Mar-2018.)
Assertion
Ref Expression
axbnd (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))

Proof of Theorem axbnd
StepHypRef Expression
1 nfnae 2441 . . . . . 6 𝑥 ¬ ∀𝑧 𝑧 = 𝑥
2 nfnae 2441 . . . . . 6 𝑥 ¬ ∀𝑧 𝑧 = 𝑦
31, 2nfan 1999 . . . . 5 𝑥(¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦)
4 nfnae 2441 . . . . . . 7 𝑧 ¬ ∀𝑧 𝑧 = 𝑥
5 nfnae 2441 . . . . . . 7 𝑧 ¬ ∀𝑧 𝑧 = 𝑦
64, 5nfan 1999 . . . . . 6 𝑧(¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦)
7 axc9 2389 . . . . . . 7 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
87imp 396 . . . . . 6 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
96, 8alrimi 2248 . . . . 5 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
103, 9alrimi 2248 . . . 4 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → ∀𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
1110ex 402 . . 3 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → ∀𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
1211orrd 890 . 2 (¬ ∀𝑧 𝑧 = 𝑥 → (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
1312orri 889 1 (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 385   ∨ wo 874  ∀wal 1651 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377 This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator