![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > axbnd | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
axbnd | ⊢ (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfnae 2441 | . . . . . 6 ⊢ Ⅎ𝑥 ¬ ∀𝑧 𝑧 = 𝑥 | |
2 | nfnae 2441 | . . . . . 6 ⊢ Ⅎ𝑥 ¬ ∀𝑧 𝑧 = 𝑦 | |
3 | 1, 2 | nfan 1999 | . . . . 5 ⊢ Ⅎ𝑥(¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) |
4 | nfnae 2441 | . . . . . . 7 ⊢ Ⅎ𝑧 ¬ ∀𝑧 𝑧 = 𝑥 | |
5 | nfnae 2441 | . . . . . . 7 ⊢ Ⅎ𝑧 ¬ ∀𝑧 𝑧 = 𝑦 | |
6 | 4, 5 | nfan 1999 | . . . . . 6 ⊢ Ⅎ𝑧(¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) |
7 | axc9 2389 | . . . . . . 7 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) | |
8 | 7 | imp 396 | . . . . . 6 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) |
9 | 6, 8 | alrimi 2248 | . . . . 5 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) |
10 | 3, 9 | alrimi 2248 | . . . 4 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) |
11 | 10 | ex 402 | . . 3 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) |
12 | 11 | orrd 890 | . 2 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) |
13 | 12 | orri 889 | 1 ⊢ (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥∀𝑧(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 |