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Mirrors > Home > MPE Home > Th. List > mo2 | Structured version Visualization version GIF version |
Description: Alternate definition of "at most one." (Contributed by NM, 8-Mar-1995.) Restrict dummy variable z. (Revised by Wolf Lammen, 28-May-2019.) |
Ref | Expression |
---|---|
mo2.1 | ⊢ Ⅎ𝑦𝜑 |
Ref | Expression |
---|---|
mo2 | ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mo2v 2477 | . 2 ⊢ (∃*𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧)) | |
2 | mo2.1 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
3 | nfv 1843 | . . . . 5 ⊢ Ⅎ𝑦 𝑥 = 𝑧 | |
4 | 2, 3 | nfim 1825 | . . . 4 ⊢ Ⅎ𝑦(𝜑 → 𝑥 = 𝑧) |
5 | 4 | nfal 2153 | . . 3 ⊢ Ⅎ𝑦∀𝑥(𝜑 → 𝑥 = 𝑧) |
6 | nfv 1843 | . . 3 ⊢ Ⅎ𝑧∀𝑥(𝜑 → 𝑥 = 𝑦) | |
7 | equequ2 1953 | . . . . 5 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦)) | |
8 | 7 | imbi2d 330 | . . . 4 ⊢ (𝑧 = 𝑦 → ((𝜑 → 𝑥 = 𝑧) ↔ (𝜑 → 𝑥 = 𝑦))) |
9 | 8 | albidv 1849 | . . 3 ⊢ (𝑧 = 𝑦 → (∀𝑥(𝜑 → 𝑥 = 𝑧) ↔ ∀𝑥(𝜑 → 𝑥 = 𝑦))) |
10 | 5, 6, 9 | cbvex 2272 | . 2 ⊢ (∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧) ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |
11 | 1, 10 | bitri 264 | 1 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∀wal 1481 ∃wex 1704 Ⅎwnf 1708 ∃*wmo 2471 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ex 1705 df-nf 1710 df-eu 2474 df-mo 2475 |
This theorem is referenced by: mo3 2507 mo 2508 rmo2 3526 nmo 29325 bj-eu3f 32829 bj-mo3OLD 32832 dffun3f 42429 |
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