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Mirrors > Home > ILE Home > Th. List > ax-9 | GIF version |
Description: Derive ax-9 1421 from ax-i9 1420, the modified version for intuitionistic logic. Although ax-9 1421 does hold intuistionistically, in intuitionistic logic it is weaker than ax-i9 1420. (Contributed by NM, 3-Feb-2015.) |
Ref | Expression |
---|---|
ax-9 | ⊢ ¬ ∀x ¬ x = y |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-i9 1420 | . . 3 ⊢ ∃x x = y | |
2 | 1 | notnoti 573 | . 2 ⊢ ¬ ¬ ∃x x = y |
3 | alnex 1385 | . 2 ⊢ (∀x ¬ x = y ↔ ¬ ∃x x = y) | |
4 | 2, 3 | mtbir 595 | 1 ⊢ ¬ ∀x ¬ x = y |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∀wal 1240 = wceq 1242 ∃wex 1378 |
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 1333 ax-gen 1335 ax-ie2 1380 ax-i9 1420 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-fal 1248 |
This theorem is referenced by: equidqe 1422 |
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