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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 2833 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐴)) | |
2 | 1 | imbi2d 343 | . . . . 5 ⊢ (𝑦 = 𝐴 → ((𝜑 → 𝑥 = 𝑦) ↔ (𝜑 → 𝑥 = 𝐴))) |
3 | 2 | albidv 1917 | . . . 4 ⊢ (𝑦 = 𝐴 → (∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥(𝜑 → 𝑥 = 𝐴))) |
4 | 3 | imbi1d 344 | . . 3 ⊢ (𝑦 = 𝐴 → ((∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃*𝑥𝜑) ↔ (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∃*𝑥𝜑))) |
5 | 19.8a 2175 | . . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) | |
6 | df-mo 2618 | . . . 4 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) | |
7 | 5, 6 | sylibr 236 | . . 3 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃*𝑥𝜑) |
8 | 4, 7 | vtoclg 3568 | . 2 ⊢ (𝐴 ∈ V → (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∃*𝑥𝜑)) |
9 | eqvisset 3512 | . . . . . 6 ⊢ (𝑥 = 𝐴 → 𝐴 ∈ V) | |
10 | 9 | imim2i 16 | . . . . 5 ⊢ ((𝜑 → 𝑥 = 𝐴) → (𝜑 → 𝐴 ∈ V)) |
11 | 10 | con3rr3 158 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ((𝜑 → 𝑥 = 𝐴) → ¬ 𝜑)) |
12 | 11 | alimdv 1913 | . . 3 ⊢ (¬ 𝐴 ∈ V → (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∀𝑥 ¬ 𝜑)) |
13 | alnex 1778 | . . . 4 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑) | |
14 | nexmo 2619 | . . . 4 ⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑) | |
15 | 13, 14 | sylbi 219 | . . 3 ⊢ (∀𝑥 ¬ 𝜑 → ∃*𝑥𝜑) |
16 | 12, 15 | syl6 35 | . 2 ⊢ (¬ 𝐴 ∈ V → (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∃*𝑥𝜑)) |
17 | 8, 16 | pm2.61i 184 | 1 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∃*𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1531 = wceq 1533 ∃wex 1776 ∈ wcel 2110 ∃*wmo 2616 Vcvv 3495 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-clab 2800 df-cleq 2814 df-clel 2893 df-v 3497 |
This theorem is referenced by: invdisj 5043 reusv1 5290 reusv2lem1 5291 opabiotafun 6739 fseqenlem2 9445 dfac2b 9550 imasaddfnlem 16795 imasvscafn 16804 bnj149 32142 |
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