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Mirrors > Home > MPE Home > Th. List > eunex | Structured version Visualization version GIF version |
Description: Existential uniqueness implies there is a value for which the wff argument is false. (Contributed by NM, 24-Oct-2010.) (Proof shortened by BJ, 2-Jan-2023.) |
Ref | Expression |
---|---|
eunex | ⊢ (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dtru 5274 | . . . 4 ⊢ ¬ ∀𝑥 𝑥 = 𝑦 | |
2 | albi 1818 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (∀𝑥𝜑 ↔ ∀𝑥 𝑥 = 𝑦)) | |
3 | 1, 2 | mtbiri 329 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ¬ ∀𝑥𝜑) |
4 | 3 | exlimiv 1930 | . 2 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ¬ ∀𝑥𝜑) |
5 | eu6 2658 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | |
6 | exnal 1826 | . 2 ⊢ (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑) | |
7 | 4, 5, 6 | 3imtr4i 294 | 1 ⊢ (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∀wal 1534 ∃wex 1779 ∃!weu 2652 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-12 2176 ax-nul 5213 ax-pow 5269 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-mo 2621 df-eu 2653 |
This theorem is referenced by: reusv2lem2 5303 neutru 33759 amosym1 33778 alneu 43330 |
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