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Mirrors > Home > MPE Home > Th. List > nexmo | Structured version Visualization version GIF version |
Description: Nonexistence implies uniqueness. (Contributed by BJ, 30-Sep-2022.) Avoid ax-11 2158. (Revised by Wolf Lammen, 16-Oct-2022.) |
Ref | Expression |
---|---|
nexmo | ⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.21 123 | . . . . 5 ⊢ (¬ 𝜑 → (𝜑 → 𝑥 = 𝑦)) | |
2 | 1 | alimi 1819 | . . . 4 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑 → 𝑥 = 𝑦)) |
3 | 2 | alrimiv 1935 | . . 3 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |
4 | 3 | 19.2d 1986 | . 2 ⊢ (∀𝑥 ¬ 𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |
5 | alnex 1789 | . . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑) | |
6 | 5 | bicomi 227 | . 2 ⊢ (¬ ∃𝑥𝜑 ↔ ∀𝑥 ¬ 𝜑) |
7 | df-mo 2539 | . 2 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) | |
8 | 4, 6, 7 | 3imtr4i 295 | 1 ⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1541 ∃wex 1787 ∃*wmo 2537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 |
This theorem depends on definitions: df-bi 210 df-ex 1788 df-mo 2539 |
This theorem is referenced by: exmo 2541 moabs 2542 exmoeu 2580 moanimlem 2619 moexexlem 2627 mo2icl 3627 mosubopt 5393 dff3 6919 disjALTV0 36599 |
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