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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 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | nfri 1530 | . . 3 ⊢ (𝜑 → ∀𝑦𝜑) |
3 | 2 | mo3h 2091 | . 2 ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) |
4 | 1 | modc 2081 | . 2 ⊢ (DECID ∃𝑥𝜑 → (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))) |
5 | 3, 4 | bitr4id 199 | 1 ⊢ (DECID ∃𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 DECID wdc 835 ∀wal 1362 Ⅎwnf 1471 ∃wex 1503 [wsb 1773 ∃*wmo 2039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 |
This theorem is referenced by: (None) |
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