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| Description: Version of eu6 2574 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 2377. (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 2574 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) | |
| 2 | euf.1 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
| 3 | nfv 1914 | . . . . 5 ⊢ Ⅎ𝑦 𝑥 = 𝑧 | |
| 4 | 2, 3 | nfbi 1903 | . . . 4 ⊢ Ⅎ𝑦(𝜑 ↔ 𝑥 = 𝑧) | 
| 5 | 4 | nfal 2323 | . . 3 ⊢ Ⅎ𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑧) | 
| 6 | nfv 1914 | . . 3 ⊢ Ⅎ𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑦) | |
| 7 | equequ2 2025 | . . . . 5 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦)) | |
| 8 | 7 | bibi2d 342 | . . . 4 ⊢ (𝑧 = 𝑦 → ((𝜑 ↔ 𝑥 = 𝑧) ↔ (𝜑 ↔ 𝑥 = 𝑦))) | 
| 9 | 8 | albidv 1920 | . . 3 ⊢ (𝑧 = 𝑦 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))) | 
| 10 | 5, 6, 9 | cbvexv1 2344 | . 2 ⊢ (∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | 
| 11 | 1, 10 | bitri 275 | 1 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∀wal 1538 ∃wex 1779 Ⅎwnf 1783 ∃!weu 2568 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-10 2141 ax-11 2157 ax-12 2177 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-mo 2540 df-eu 2569 | 
| This theorem is referenced by: eu1 2610 | 
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