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Mirrors > Home > ILE Home > Th. List > mo3 | GIF version |
Description: Alternate definition of "at most one". Definition of [BellMachover] p. 460, except that definition has the side condition that 𝑦 not occur in 𝜑 in place of our hypothesis. (Contributed by NM, 8-Mar-1995.) |
Ref | Expression |
---|---|
mo3.1 | ⊢ Ⅎ𝑦𝜑 |
Ref | Expression |
---|---|
mo3 | ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mo3.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | nfri 1512 | . 2 ⊢ (𝜑 → ∀𝑦𝜑) |
3 | 2 | mo3h 2072 | 1 ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1346 Ⅎwnf 1453 [wsb 1755 ∃*wmo 2020 |
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-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 |
This theorem depends on definitions: df-bi 116 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 |
This theorem is referenced by: sbmo 2078 rmo3f 2927 rmo3 3046 isarep2 5285 |
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