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Mirrors > Home > ILE Home > Th. List > alexim | GIF version |
Description: One direction of theorem 19.6 of [Margaris] p. 89. The converse holds given a decidability condition, as seen at alexdc 1598. (Contributed by Jim Kingdon, 2-Jul-2018.) |
Ref | Expression |
---|---|
alexim | ⊢ (∀𝑥𝜑 → ¬ ∃𝑥 ¬ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.24 610 | . . . . 5 ⊢ (𝜑 → (¬ 𝜑 → ⊥)) | |
2 | 1 | alimi 1431 | . . . 4 ⊢ (∀𝑥𝜑 → ∀𝑥(¬ 𝜑 → ⊥)) |
3 | exim 1578 | . . . 4 ⊢ (∀𝑥(¬ 𝜑 → ⊥) → (∃𝑥 ¬ 𝜑 → ∃𝑥⊥)) | |
4 | 2, 3 | syl 14 | . . 3 ⊢ (∀𝑥𝜑 → (∃𝑥 ¬ 𝜑 → ∃𝑥⊥)) |
5 | nfv 1508 | . . . 4 ⊢ Ⅎ𝑥⊥ | |
6 | 5 | 19.9 1623 | . . 3 ⊢ (∃𝑥⊥ ↔ ⊥) |
7 | 4, 6 | syl6ib 160 | . 2 ⊢ (∀𝑥𝜑 → (∃𝑥 ¬ 𝜑 → ⊥)) |
8 | dfnot 1349 | . 2 ⊢ (¬ ∃𝑥 ¬ 𝜑 ↔ (∃𝑥 ¬ 𝜑 → ⊥)) | |
9 | 7, 8 | sylibr 133 | 1 ⊢ (∀𝑥𝜑 → ¬ ∃𝑥 ¬ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1329 ⊥wfal 1336 ∃wex 1468 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-5 1423 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-4 1487 ax-17 1506 ax-ial 1514 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-fal 1337 df-nf 1437 |
This theorem is referenced by: exnalim 1625 exists2 2096 |
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