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| Mirrors > Home > MPE Home > Th. List > Mathboxes > moeu2 | Structured version Visualization version GIF version | ||
| Description: Uniqueness is equivalent to non-existence or unique existence. Alternate definition of the at-most-one quantifier, in terms of the existential quantifier and the unique existential quantifier. (Contributed by Peter Mazsa, 19-Nov-2024.) | 
| Ref | Expression | 
|---|---|
| moeu2 | ⊢ (∃*𝑥𝜑 ↔ (¬ ∃𝑥𝜑 ∨ ∃!𝑥𝜑)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | moeu 2583 | . 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) | |
| 2 | imor 854 | . 2 ⊢ ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (¬ ∃𝑥𝜑 ∨ ∃!𝑥𝜑)) | |
| 3 | 1, 2 | bitri 275 | 1 ⊢ (∃*𝑥𝜑 ↔ (¬ ∃𝑥𝜑 ∨ ∃!𝑥𝜑)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∨ wo 848 ∃wex 1779 ∃*wmo 2538 ∃!weu 2568 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-mo 2540 df-eu 2569 | 
| This theorem is referenced by: mopickr 38364 | 
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