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Mirrors > Home > ILE Home > Th. List > ax-9 | GIF version |
Description: Derive ax-9 1470 from ax-i9 1469, the modified version for intuitionistic logic. Although ax-9 1470 does hold intuistionistically, in intuitionistic logic it is weaker than ax-i9 1469. (Contributed by NM, 3-Feb-2015.) |
Ref | Expression |
---|---|
ax-9 | ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-i9 1469 | . . 3 ⊢ ∃𝑥 𝑥 = 𝑦 | |
2 | 1 | notnoti 610 | . 2 ⊢ ¬ ¬ ∃𝑥 𝑥 = 𝑦 |
3 | alnex 1434 | . 2 ⊢ (∀𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∃𝑥 𝑥 = 𝑦) | |
4 | 2, 3 | mtbir 632 | 1 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∀wal 1288 = wceq 1290 ∃wex 1427 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-5 1382 ax-gen 1384 ax-ie2 1429 ax-i9 1469 |
This theorem depends on definitions: df-bi 116 df-tru 1293 df-fal 1296 |
This theorem is referenced by: equidqe 1471 |
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