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Mirrors > Home > MPE Home > Th. List > euf | Structured version Visualization version GIF version |
Description: Version of eu6 2645 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 2389. (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 2645 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) | |
2 | euf.1 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
3 | nfv 2013 | . . . . 5 ⊢ Ⅎ𝑦 𝑥 = 𝑧 | |
4 | 2, 3 | nfbi 2006 | . . . 4 ⊢ Ⅎ𝑦(𝜑 ↔ 𝑥 = 𝑧) |
5 | 4 | nfal 2355 | . . 3 ⊢ Ⅎ𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑧) |
6 | nfv 2013 | . . 3 ⊢ Ⅎ𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑦) | |
7 | equequ2 2130 | . . . . 5 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦)) | |
8 | 7 | bibi2d 334 | . . . 4 ⊢ (𝑧 = 𝑦 → ((𝜑 ↔ 𝑥 = 𝑧) ↔ (𝜑 ↔ 𝑥 = 𝑦))) |
9 | 8 | albidv 2019 | . . 3 ⊢ (𝑧 = 𝑦 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))) |
10 | 5, 6, 9 | cbvexv1 2368 | . 2 ⊢ (∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) |
11 | 1, 10 | bitri 267 | 1 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∀wal 1654 ∃wex 1878 Ⅎwnf 1882 ∃!weu 2639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-10 2192 ax-11 2207 ax-12 2220 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-tru 1660 df-ex 1879 df-nf 1883 df-mo 2605 df-eu 2640 |
This theorem is referenced by: eu1 2695 eu1OLD 2696 |
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