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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 2621 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐴)) | |
2 | 1 | imbi2d 329 | . . . . 5 ⊢ (𝑦 = 𝐴 → ((𝜑 → 𝑥 = 𝑦) ↔ (𝜑 → 𝑥 = 𝐴))) |
3 | 2 | albidv 1836 | . . . 4 ⊢ (𝑦 = 𝐴 → (∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥(𝜑 → 𝑥 = 𝐴))) |
4 | 3 | imbi1d 330 | . . 3 ⊢ (𝑦 = 𝐴 → ((∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃*𝑥𝜑) ↔ (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∃*𝑥𝜑))) |
5 | 19.8a 2039 | . . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) | |
6 | mo2v 2465 | . . . 4 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) | |
7 | 5, 6 | sylibr 223 | . . 3 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃*𝑥𝜑) |
8 | 4, 7 | vtoclg 3239 | . 2 ⊢ (𝐴 ∈ V → (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∃*𝑥𝜑)) |
9 | eqvisset 3184 | . . . . . 6 ⊢ (𝑥 = 𝐴 → 𝐴 ∈ V) | |
10 | 9 | imim2i 16 | . . . . 5 ⊢ ((𝜑 → 𝑥 = 𝐴) → (𝜑 → 𝐴 ∈ V)) |
11 | 10 | con3rr3 150 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ((𝜑 → 𝑥 = 𝐴) → ¬ 𝜑)) |
12 | 11 | alimdv 1832 | . . 3 ⊢ (¬ 𝐴 ∈ V → (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∀𝑥 ¬ 𝜑)) |
13 | alnex 1697 | . . . 4 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑) | |
14 | exmo 2483 | . . . . 5 ⊢ (∃𝑥𝜑 ∨ ∃*𝑥𝜑) | |
15 | 14 | ori 389 | . . . 4 ⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑) |
16 | 13, 15 | sylbi 206 | . . 3 ⊢ (∀𝑥 ¬ 𝜑 → ∃*𝑥𝜑) |
17 | 12, 16 | syl6 34 | . 2 ⊢ (¬ 𝐴 ∈ V → (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∃*𝑥𝜑)) |
18 | 8, 17 | pm2.61i 175 | 1 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∃*𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1473 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ∃*wmo 2459 Vcvv 3173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-12 2034 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-v 3175 |
This theorem is referenced by: invdisj 4571 reusv1 4792 reusv2lem1 4794 opabiotafun 6169 fseqenlem2 8731 dfac2 8836 imasaddfnlem 16011 imasvscafn 16020 bnj149 30199 |
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