![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > ax-12 | GIF version |
Description: Rederive the original version of the axiom from ax-i12 1395. (Contributed by Mario Carneiro, 3-Feb-2015.) |
Ref | Expression |
---|---|
ax-12 | ⊢ (¬ ∀z z = x → (¬ ∀z z = y → (x = y → ∀z x = y))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-i12 1395 | . . . 4 ⊢ (∀z z = x ∨ (∀z z = y ∨ ∀z(x = y → ∀z x = y))) | |
2 | 1 | ori 641 | . . 3 ⊢ (¬ ∀z z = x → (∀z z = y ∨ ∀z(x = y → ∀z x = y))) |
3 | 2 | ord 642 | . 2 ⊢ (¬ ∀z z = x → (¬ ∀z z = y → ∀z(x = y → ∀z x = y))) |
4 | ax-4 1397 | . 2 ⊢ (∀z(x = y → ∀z x = y) → (x = y → ∀z x = y)) | |
5 | 3, 4 | syl6 29 | 1 ⊢ (¬ ∀z z = x → (¬ ∀z z = y → (x = y → ∀z x = y))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 628 ∀wal 1240 = wceq 1242 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in2 545 ax-io 629 ax-i12 1395 ax-4 1397 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |