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Mirrors > Home > ILE Home > Th. List > eu5 | GIF version |
Description: Uniqueness in terms of "at most one". (Contributed by NM, 23-Mar-1995.) (Proof rewritten by Jim Kingdon, 27-May-2018.) |
Ref | Expression |
---|---|
eu5 | ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | euex 2056 | . . 3 ⊢ (∃!𝑥𝜑 → ∃𝑥𝜑) | |
2 | eumo 2058 | . . 3 ⊢ (∃!𝑥𝜑 → ∃*𝑥𝜑) | |
3 | 1, 2 | jca 306 | . 2 ⊢ (∃!𝑥𝜑 → (∃𝑥𝜑 ∧ ∃*𝑥𝜑)) |
4 | df-mo 2030 | . . . . 5 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) | |
5 | 4 | biimpi 120 | . . . 4 ⊢ (∃*𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑)) |
6 | 5 | imp 124 | . . 3 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥𝜑) → ∃!𝑥𝜑) |
7 | 6 | ancoms 268 | . 2 ⊢ ((∃𝑥𝜑 ∧ ∃*𝑥𝜑) → ∃!𝑥𝜑) |
8 | 3, 7 | impbii 126 | 1 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∃wex 1492 ∃!weu 2026 ∃*wmo 2027 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 |
This theorem is referenced by: exmoeu2 2074 euan 2082 eu4 2088 euim 2094 euexex 2111 2euex 2113 2euswapdc 2117 2exeu 2118 reu5 2690 reuss2 3416 funcnv3 5279 fnres 5333 fnopabg 5340 brprcneu 5509 dff3im 5662 recmulnqg 7390 uptx 13777 |
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