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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 2157. (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 1808 | . . . 4 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑 → 𝑥 = 𝑦)) |
3 | 2 | alrimiv 1924 | . . 3 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |
4 | 3 | 19.2d 1978 | . 2 ⊢ (∀𝑥 ¬ 𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |
5 | alnex 1778 | . . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑) | |
6 | 5 | bicomi 226 | . 2 ⊢ (¬ ∃𝑥𝜑 ↔ ∀𝑥 ¬ 𝜑) |
7 | df-mo 2618 | . 2 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) | |
8 | 4, 6, 7 | 3imtr4i 294 | 1 ⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1531 ∃wex 1776 ∃*wmo 2616 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 |
This theorem depends on definitions: df-bi 209 df-ex 1777 df-mo 2618 |
This theorem is referenced by: exmo 2620 moabs 2621 exmoeu 2662 moanimlem 2699 moexexlem 2707 mo2icl 3704 mosubopt 5392 dff3 6860 disjALTV0 35978 |
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