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Mirrors > Home > MPE Home > Th. List > mo2icl | Structured version Visualization version GIF version |
Description: Theorem for inferring "at most one." (Contributed by NM, 17-Oct-1996.) |
Ref | Expression |
---|---|
mo2icl | ⊢ (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∃*𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq2 2633 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐴)) | |
2 | 1 | imbi2d 330 | . . . . 5 ⊢ (𝑦 = 𝐴 → ((𝜑 → 𝑥 = 𝑦) ↔ (𝜑 → 𝑥 = 𝐴))) |
3 | 2 | albidv 1849 | . . . 4 ⊢ (𝑦 = 𝐴 → (∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥(𝜑 → 𝑥 = 𝐴))) |
4 | 3 | imbi1d 331 | . . 3 ⊢ (𝑦 = 𝐴 → ((∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃*𝑥𝜑) ↔ (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∃*𝑥𝜑))) |
5 | 19.8a 2052 | . . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) | |
6 | mo2v 2477 | . . . 4 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) | |
7 | 5, 6 | sylibr 224 | . . 3 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃*𝑥𝜑) |
8 | 4, 7 | vtoclg 3266 | . 2 ⊢ (𝐴 ∈ V → (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∃*𝑥𝜑)) |
9 | eqvisset 3211 | . . . . . 6 ⊢ (𝑥 = 𝐴 → 𝐴 ∈ V) | |
10 | 9 | imim2i 16 | . . . . 5 ⊢ ((𝜑 → 𝑥 = 𝐴) → (𝜑 → 𝐴 ∈ V)) |
11 | 10 | con3rr3 151 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ((𝜑 → 𝑥 = 𝐴) → ¬ 𝜑)) |
12 | 11 | alimdv 1845 | . . 3 ⊢ (¬ 𝐴 ∈ V → (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∀𝑥 ¬ 𝜑)) |
13 | alnex 1706 | . . . 4 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑) | |
14 | exmo 2495 | . . . . 5 ⊢ (∃𝑥𝜑 ∨ ∃*𝑥𝜑) | |
15 | 14 | ori 390 | . . . 4 ⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑) |
16 | 13, 15 | sylbi 207 | . . 3 ⊢ (∀𝑥 ¬ 𝜑 → ∃*𝑥𝜑) |
17 | 12, 16 | syl6 35 | . 2 ⊢ (¬ 𝐴 ∈ V → (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∃*𝑥𝜑)) |
18 | 8, 17 | pm2.61i 176 | 1 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∃*𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1481 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ∃*wmo 2471 Vcvv 3200 |
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-9 1999 ax-10 2019 ax-12 2047 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-v 3202 |
This theorem is referenced by: invdisj 4638 reusv1 4866 reusv2lem1 4868 opabiotafun 6259 fseqenlem2 8848 dfac2 8953 imasaddfnlem 16188 imasvscafn 16197 bnj149 30945 |
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