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Mirrors > Home > ILE Home > Th. List > eu3h | GIF version |
Description: An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) (New usage is discouraged.) |
Ref | Expression |
---|---|
eu3h.1 | ⊢ (𝜑 → ∀𝑦𝜑) |
Ref | Expression |
---|---|
eu3h | ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | euex 2056 | . . 3 ⊢ (∃!𝑥𝜑 → ∃𝑥𝜑) | |
2 | eu3h.1 | . . . 4 ⊢ (𝜑 → ∀𝑦𝜑) | |
3 | 2 | eumo0 2057 | . . 3 ⊢ (∃!𝑥𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |
4 | 1, 3 | jca 306 | . 2 ⊢ (∃!𝑥𝜑 → (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) |
5 | 2 | nfi 1462 | . . . . 5 ⊢ Ⅎ𝑦𝜑 |
6 | 5 | mo23 2067 | . . . 4 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) |
7 | 6 | anim2i 342 | . . 3 ⊢ ((∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) → (∃𝑥𝜑 ∧ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))) |
8 | 5 | eu2 2070 | . . 3 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))) |
9 | 7, 8 | sylibr 134 | . 2 ⊢ ((∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) → ∃!𝑥𝜑) |
10 | 4, 9 | impbii 126 | 1 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∀wal 1351 ∃wex 1492 [wsb 1762 ∃!weu 2026 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-eu 2029 |
This theorem is referenced by: eu3 2072 mo2r 2078 2eu4 2119 |
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