Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > ax-9 | GIF version |
Description: Derive ax-9 1424 from ax-i9 1423, the modified version for intuitionistic logic. Although ax-9 1424 does hold intuistionistically, in intuitionistic logic it is weaker than ax-i9 1423. (Contributed by NM, 3-Feb-2015.) |
Ref | Expression |
---|---|
ax-9 | ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-i9 1423 | . . 3 ⊢ ∃𝑥 𝑥 = 𝑦 | |
2 | 1 | notnoti 574 | . 2 ⊢ ¬ ¬ ∃𝑥 𝑥 = 𝑦 |
3 | alnex 1388 | . 2 ⊢ (∀𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∃𝑥 𝑥 = 𝑦) | |
4 | 2, 3 | mtbir 596 | 1 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∀wal 1241 = wceq 1243 ∃wex 1381 |
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-in1 544 ax-in2 545 ax-5 1336 ax-gen 1338 ax-ie2 1383 ax-i9 1423 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-fal 1249 |
This theorem is referenced by: equidqe 1425 |
Copyright terms: Public domain | W3C validator |