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Mirrors > Home > MPE Home > Th. List > Mathboxes > nexmo1 | Structured version Visualization version GIF version |
Description: If there is no case where wff is true, it is true for at most one case. (Contributed by Peter Mazsa, 27-Sep-2021.) |
Ref | Expression |
---|---|
nexmo1 | ⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.21 123 | . 2 ⊢ (¬ ∃𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑)) | |
2 | moeu 2583 | . 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) | |
3 | 1, 2 | sylibr 233 | 1 ⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∃wex 1782 ∃*wmo 2538 ∃!weu 2568 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1783 df-mo 2540 df-eu 2569 |
This theorem is referenced by: (None) |
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