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Mirrors > Home > MPE Home > Th. List > euf | Structured version Visualization version GIF version |
Description: Version of eu6 2652 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Wolf Lammen, 30-Oct-2018.) Avoid ax-13 2381. (Revised by Wolf Lammen, 16-Oct-2022.) |
Ref | Expression |
---|---|
euf.1 | ⊢ Ⅎ𝑦𝜑 |
Ref | Expression |
---|---|
euf | ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eu6 2652 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) | |
2 | euf.1 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
3 | nfv 1906 | . . . . 5 ⊢ Ⅎ𝑦 𝑥 = 𝑧 | |
4 | 2, 3 | nfbi 1895 | . . . 4 ⊢ Ⅎ𝑦(𝜑 ↔ 𝑥 = 𝑧) |
5 | 4 | nfal 2333 | . . 3 ⊢ Ⅎ𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑧) |
6 | nfv 1906 | . . 3 ⊢ Ⅎ𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑦) | |
7 | equequ2 2024 | . . . . 5 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦)) | |
8 | 7 | bibi2d 344 | . . . 4 ⊢ (𝑧 = 𝑦 → ((𝜑 ↔ 𝑥 = 𝑧) ↔ (𝜑 ↔ 𝑥 = 𝑦))) |
9 | 8 | albidv 1912 | . . 3 ⊢ (𝑧 = 𝑦 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))) |
10 | 5, 6, 9 | cbvexv1 2353 | . 2 ⊢ (∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) |
11 | 1, 10 | bitri 276 | 1 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∀wal 1526 ∃wex 1771 Ⅎwnf 1775 ∃!weu 2646 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-10 2136 ax-11 2151 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-mo 2615 df-eu 2647 |
This theorem is referenced by: eu1 2687 |
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