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Mirrors > Home > ILE Home > Th. List > mo2dc | GIF version |
Description: Alternate definition of "at most one" where existence is decidable. (Contributed by Jim Kingdon, 2-Jul-2018.) |
Ref | Expression |
---|---|
mo2dc.1 | ⊢ Ⅎ𝑦𝜑 |
Ref | Expression |
---|---|
mo2dc | ⊢ (DECID ∃𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mo2dc.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | modc 2040 | . 2 ⊢ (DECID ∃𝑥𝜑 → (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))) |
3 | 1 | nfri 1499 | . . 3 ⊢ (𝜑 → ∀𝑦𝜑) |
4 | 3 | mo3h 2050 | . 2 ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) |
5 | 2, 4 | syl6rbbr 198 | 1 ⊢ (DECID ∃𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 DECID wdc 819 ∀wal 1329 Ⅎwnf 1436 ∃wex 1468 [wsb 1735 ∃*wmo 1998 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 |
This theorem is referenced by: (None) |
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