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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 1529 | . 2 ⊢ (𝜑 → ∀𝑦𝜑) |
3 | 2 | mo3h 2089 | 1 ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∀wal 1361 Ⅎwnf 1470 [wsb 1772 ∃*wmo 2037 |
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-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 |
This theorem depends on definitions: df-bi 117 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 |
This theorem is referenced by: sbmo 2095 rmo3f 2946 rmo3 3066 isarep2 5315 |
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