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Mirrors > Home > MPE Home > Th. List > exists2 | Structured version Visualization version GIF version |
Description: A condition implying that at least two things exist. (Contributed by NM, 10-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Ref | Expression |
---|---|
exists2 | ⊢ ((∃𝑥𝜑 ∧ ∃𝑥 ¬ 𝜑) → ¬ ∃!𝑥 𝑥 = 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfeu1 2628 | . . . . . 6 ⊢ Ⅎ𝑥∃!𝑥 𝑥 = 𝑥 | |
2 | nfa1 2184 | . . . . . 6 ⊢ Ⅎ𝑥∀𝑥𝜑 | |
3 | exists1 2710 | . . . . . . 7 ⊢ (∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦) | |
4 | axc16 2300 | . . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑)) | |
5 | 3, 4 | sylbi 207 | . . . . . 6 ⊢ (∃!𝑥 𝑥 = 𝑥 → (𝜑 → ∀𝑥𝜑)) |
6 | 1, 2, 5 | exlimd 2243 | . . . . 5 ⊢ (∃!𝑥 𝑥 = 𝑥 → (∃𝑥𝜑 → ∀𝑥𝜑)) |
7 | 6 | com12 32 | . . . 4 ⊢ (∃𝑥𝜑 → (∃!𝑥 𝑥 = 𝑥 → ∀𝑥𝜑)) |
8 | alex 1901 | . . . 4 ⊢ (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑) | |
9 | 7, 8 | syl6ib 241 | . . 3 ⊢ (∃𝑥𝜑 → (∃!𝑥 𝑥 = 𝑥 → ¬ ∃𝑥 ¬ 𝜑)) |
10 | 9 | con2d 131 | . 2 ⊢ (∃𝑥𝜑 → (∃𝑥 ¬ 𝜑 → ¬ ∃!𝑥 𝑥 = 𝑥)) |
11 | 10 | imp 393 | 1 ⊢ ((∃𝑥𝜑 ∧ ∃𝑥 ¬ 𝜑) → ¬ ∃!𝑥 𝑥 = 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 382 ∀wal 1629 ∃wex 1852 ∃!weu 2618 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-tru 1634 df-ex 1853 df-nf 1858 df-eu 2622 |
This theorem is referenced by: (None) |
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