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Mirrors > Home > ILE Home > Th. List > exmoeudc | GIF version |
Description: Existence in terms of "at most one" and uniqueness. (Contributed by Jim Kingdon, 3-Jul-2018.) |
Ref | Expression |
---|---|
exmoeudc | ⊢ (DECID ∃𝑥𝜑 → (∃𝑥𝜑 ↔ (∃*𝑥𝜑 → ∃!𝑥𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mo 2018 | . . . 4 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) | |
2 | 1 | biimpi 119 | . . 3 ⊢ (∃*𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑)) |
3 | 2 | com12 30 | . 2 ⊢ (∃𝑥𝜑 → (∃*𝑥𝜑 → ∃!𝑥𝜑)) |
4 | 1 | biimpri 132 | . . . 4 ⊢ ((∃𝑥𝜑 → ∃!𝑥𝜑) → ∃*𝑥𝜑) |
5 | euex 2044 | . . . 4 ⊢ (∃!𝑥𝜑 → ∃𝑥𝜑) | |
6 | 4, 5 | imim12i 59 | . . 3 ⊢ ((∃*𝑥𝜑 → ∃!𝑥𝜑) → ((∃𝑥𝜑 → ∃!𝑥𝜑) → ∃𝑥𝜑)) |
7 | peircedc 904 | . . 3 ⊢ (DECID ∃𝑥𝜑 → (((∃𝑥𝜑 → ∃!𝑥𝜑) → ∃𝑥𝜑) → ∃𝑥𝜑)) | |
8 | 6, 7 | syl5 32 | . 2 ⊢ (DECID ∃𝑥𝜑 → ((∃*𝑥𝜑 → ∃!𝑥𝜑) → ∃𝑥𝜑)) |
9 | 3, 8 | impbid2 142 | 1 ⊢ (DECID ∃𝑥𝜑 → (∃𝑥𝜑 ↔ (∃*𝑥𝜑 → ∃!𝑥𝜑))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 DECID wdc 824 ∃wex 1480 ∃!weu 2014 ∃*wmo 2015 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 |
This theorem is referenced by: (None) |
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