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| Mirrors > Home > MPE Home > Th. List > mof | Structured version Visualization version GIF version | ||
| Description: Version of df-mo 2540 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by NM, 8-Mar-1995.) Extract dfmo 2596 from this proof, and prove mof 2563 from it (as of 30-Sep-2022, directly from df-mo 2540). (Revised by Wolf Lammen, 28-May-2019.) Avoid ax-13 2377. (Revised by Wolf Lammen, 16-Oct-2022.) |
| Ref | Expression |
|---|---|
| mof.1 | ⊢ Ⅎ𝑦𝜑 |
| Ref | Expression |
|---|---|
| mof | ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-mo 2540 | . 2 ⊢ (∃*𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧)) | |
| 2 | mof.1 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
| 3 | nfv 1914 | . . . . 5 ⊢ Ⅎ𝑦 𝑥 = 𝑧 | |
| 4 | 2, 3 | nfim 1896 | . . . 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 | imbi2d 340 | . . . 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: → wi 4 ↔ wb 206 ∀wal 1538 ∃wex 1779 Ⅎwnf 1783 ∃*wmo 2538 |
| 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-ex 1780 df-nf 1784 df-mo 2540 |
| This theorem is referenced by: mo3 2564 mo 2565 rmo2 3887 nmo 32509 fineqvrep 35109 bj-eu3f 36842 dffun3f 49201 |
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