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| 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 | dtruALT2 5369 | . . . 4 ⊢ ¬ ∀𝑥 𝑥 = 𝑦 | |
| 2 | albi 1817 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (∀𝑥𝜑 ↔ ∀𝑥 𝑥 = 𝑦)) | |
| 3 | 1, 2 | mtbiri 327 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ¬ ∀𝑥𝜑) | 
| 4 | 3 | exlimiv 1929 | . 2 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ¬ ∀𝑥𝜑) | 
| 5 | eu6 2573 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | |
| 6 | exnal 1826 | . 2 ⊢ (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑) | |
| 7 | 4, 5, 6 | 3imtr4i 292 | 1 ⊢ (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∀wal 1537 ∃wex 1778 ∃!weu 2567 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-12 2176 ax-nul 5305 ax-pow 5364 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1779 df-nf 1783 df-mo 2539 df-eu 2568 | 
| This theorem is referenced by: reusv2lem2 5398 neutru 36409 alneu 47141 | 
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