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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 2147. (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 1806 | . . . 4 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑 → 𝑥 = 𝑦)) |
3 | 2 | alrimiv 1923 | . . 3 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |
4 | 3 | 19.2d 1974 | . 2 ⊢ (∀𝑥 ¬ 𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |
5 | alnex 1776 | . . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑) | |
6 | 5 | bicomi 223 | . 2 ⊢ (¬ ∃𝑥𝜑 ↔ ∀𝑥 ¬ 𝜑) |
7 | df-mo 2529 | . 2 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) | |
8 | 4, 6, 7 | 3imtr4i 291 | 1 ⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1532 ∃wex 1774 ∃*wmo 2527 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 |
This theorem depends on definitions: df-bi 206 df-ex 1775 df-mo 2529 |
This theorem is referenced by: exmo 2531 moabs 2532 exmoeu 2570 moanimlem 2607 moexexlem 2615 mo2icl 3708 mosubopt 5518 dff3 7116 disjALTV0 38454 |
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