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Mirrors > Home > ILE Home > Th. List > moaneu | GIF version |
Description: Nested at-most-one and unique existential quantifiers. (Contributed by NM, 25-Jan-2006.) |
Ref | Expression |
---|---|
moaneu | ⊢ ∃*𝑥(𝜑 ∧ ∃!𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eumo 2031 | . . 3 ⊢ (∃!𝑥𝜑 → ∃*𝑥𝜑) | |
2 | nfeu1 2010 | . . . 4 ⊢ Ⅎ𝑥∃!𝑥𝜑 | |
3 | 2 | moanim 2073 | . . 3 ⊢ (∃*𝑥(∃!𝑥𝜑 ∧ 𝜑) ↔ (∃!𝑥𝜑 → ∃*𝑥𝜑)) |
4 | 1, 3 | mpbir 145 | . 2 ⊢ ∃*𝑥(∃!𝑥𝜑 ∧ 𝜑) |
5 | ancom 264 | . . 3 ⊢ ((𝜑 ∧ ∃!𝑥𝜑) ↔ (∃!𝑥𝜑 ∧ 𝜑)) | |
6 | 5 | mobii 2036 | . 2 ⊢ (∃*𝑥(𝜑 ∧ ∃!𝑥𝜑) ↔ ∃*𝑥(∃!𝑥𝜑 ∧ 𝜑)) |
7 | 4, 6 | mpbir 145 | 1 ⊢ ∃*𝑥(𝜑 ∧ ∃!𝑥𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∃!weu 1999 ∃*wmo 2000 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 |
This theorem is referenced by: (None) |
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