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Mirrors > Home > ILE Home > Th. List > moaneu | GIF version |
Description: Nested "at most one" and uniqueness quantifiers. (Contributed by NM, 25-Jan-2006.) |
Ref | Expression |
---|---|
moaneu | ⊢ ∃*x(φ ∧ ∃!xφ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eumo 1929 | . . 3 ⊢ (∃!xφ → ∃*xφ) | |
2 | nfeu1 1908 | . . . 4 ⊢ Ⅎx∃!xφ | |
3 | 2 | moanim 1971 | . . 3 ⊢ (∃*x(∃!xφ ∧ φ) ↔ (∃!xφ → ∃*xφ)) |
4 | 1, 3 | mpbir 134 | . 2 ⊢ ∃*x(∃!xφ ∧ φ) |
5 | ancom 253 | . . 3 ⊢ ((φ ∧ ∃!xφ) ↔ (∃!xφ ∧ φ)) | |
6 | 5 | mobii 1934 | . 2 ⊢ (∃*x(φ ∧ ∃!xφ) ↔ ∃*x(∃!xφ ∧ φ)) |
7 | 4, 6 | mpbir 134 | 1 ⊢ ∃*x(φ ∧ ∃!xφ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∃!weu 1897 ∃*wmo 1898 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 |
This theorem is referenced by: (None) |
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