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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.) |
Ref | Expression |
---|---|
eunex | ⊢ (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dtru 4783 | . . . . 5 ⊢ ¬ ∀𝑥 𝑥 = 𝑦 | |
2 | alim 1729 | . . . . 5 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → (∀𝑥𝜑 → ∀𝑥 𝑥 = 𝑦)) | |
3 | 1, 2 | mtoi 189 | . . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → ¬ ∀𝑥𝜑) |
4 | 3 | exlimiv 1845 | . . 3 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → ¬ ∀𝑥𝜑) |
5 | 4 | adantl 481 | . 2 ⊢ ((∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) → ¬ ∀𝑥𝜑) |
6 | eu3v 2486 | . 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) | |
7 | exnal 1744 | . 2 ⊢ (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑) | |
8 | 5, 6, 7 | 3imtr4i 280 | 1 ⊢ (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∀wal 1473 ∃wex 1695 ∃!weu 2458 |
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-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-nul 4717 ax-pow 4769 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-eu 2462 df-mo 2463 |
This theorem is referenced by: reusv2lem2 4795 reusv2lem2OLD 4796 unnt 31577 amosym1 31595 alneu 39850 |
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