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Mirrors > Home > ILE Home > Th. List > euf | GIF version |
Description: A version of the existential uniqueness definition with a hypothesis instead of a distinct variable condition. (Contributed by NM, 12-Aug-1993.) |
Ref | Expression |
---|---|
euf.1 | ⊢ (𝜑 → ∀𝑦𝜑) |
Ref | Expression |
---|---|
euf | ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-eu 1980 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) | |
2 | euf.1 | . . . . 5 ⊢ (𝜑 → ∀𝑦𝜑) | |
3 | ax-17 1491 | . . . . 5 ⊢ (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧) | |
4 | 2, 3 | hbbi 1512 | . . . 4 ⊢ ((𝜑 ↔ 𝑥 = 𝑧) → ∀𝑦(𝜑 ↔ 𝑥 = 𝑧)) |
5 | 4 | hbal 1438 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∀𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) |
6 | ax-17 1491 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∀𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | |
7 | equequ2 1674 | . . . . 5 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦)) | |
8 | 7 | bibi2d 231 | . . . 4 ⊢ (𝑧 = 𝑦 → ((𝜑 ↔ 𝑥 = 𝑧) ↔ (𝜑 ↔ 𝑥 = 𝑦))) |
9 | 8 | albidv 1780 | . . 3 ⊢ (𝑧 = 𝑦 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))) |
10 | 5, 6, 9 | cbvexh 1713 | . 2 ⊢ (∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) |
11 | 1, 10 | bitri 183 | 1 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1314 ∃wex 1453 ∃!weu 1977 |
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-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 |
This theorem depends on definitions: df-bi 116 df-eu 1980 |
This theorem is referenced by: eu1 2002 eumo0 2008 |
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