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Mirrors > Home > MPE Home > Th. List > exmoeu | Structured version Visualization version GIF version |
Description: Existence is equivalent to uniqueness implying existential uniqueness. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Wolf Lammen, 5-Dec-2018.) (Proof shortened by BJ, 7-Oct-2022.) |
Ref | Expression |
---|---|
exmoeu | ⊢ (∃𝑥𝜑 ↔ (∃*𝑥𝜑 → ∃!𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exmoeub 2579 | . . 3 ⊢ (∃𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃!𝑥𝜑)) | |
2 | 1 | biimpd 232 | . 2 ⊢ (∃𝑥𝜑 → (∃*𝑥𝜑 → ∃!𝑥𝜑)) |
3 | nexmo 2540 | . . . 4 ⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑) | |
4 | 3 | con1i 149 | . . 3 ⊢ (¬ ∃*𝑥𝜑 → ∃𝑥𝜑) |
5 | euex 2576 | . . 3 ⊢ (∃!𝑥𝜑 → ∃𝑥𝜑) | |
6 | 4, 5 | ja 189 | . 2 ⊢ ((∃*𝑥𝜑 → ∃!𝑥𝜑) → ∃𝑥𝜑) |
7 | 2, 6 | impbii 212 | 1 ⊢ (∃𝑥𝜑 ↔ (∃*𝑥𝜑 → ∃!𝑥𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∃wex 1787 ∃*wmo 2537 ∃!weu 2567 |
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-an 400 df-ex 1788 df-mo 2539 df-eu 2568 |
This theorem is referenced by: (None) |
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