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Mirrors > Home > ILE Home > Th. List > 2eu7 | GIF version |
Description: Two equivalent expressions for double existential uniqueness. (Contributed by NM, 19-Feb-2005.) |
Ref | Expression |
---|---|
2eu7 | ⊢ ((∃!x∃yφ ∧ ∃!y∃xφ) ↔ ∃!x∃!y(∃xφ ∧ ∃yφ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbe1 1381 | . . . 4 ⊢ (∃xφ → ∀x∃xφ) | |
2 | 1 | hbeu 1918 | . . 3 ⊢ (∃!y∃xφ → ∀x∃!y∃xφ) |
3 | 2 | euan 1953 | . 2 ⊢ (∃!x(∃!y∃xφ ∧ ∃yφ) ↔ (∃!y∃xφ ∧ ∃!x∃yφ)) |
4 | ancom 253 | . . . . 5 ⊢ ((∃xφ ∧ ∃yφ) ↔ (∃yφ ∧ ∃xφ)) | |
5 | 4 | eubii 1906 | . . . 4 ⊢ (∃!y(∃xφ ∧ ∃yφ) ↔ ∃!y(∃yφ ∧ ∃xφ)) |
6 | hbe1 1381 | . . . . 5 ⊢ (∃yφ → ∀y∃yφ) | |
7 | 6 | euan 1953 | . . . 4 ⊢ (∃!y(∃yφ ∧ ∃xφ) ↔ (∃yφ ∧ ∃!y∃xφ)) |
8 | ancom 253 | . . . 4 ⊢ ((∃yφ ∧ ∃!y∃xφ) ↔ (∃!y∃xφ ∧ ∃yφ)) | |
9 | 5, 7, 8 | 3bitri 195 | . . 3 ⊢ (∃!y(∃xφ ∧ ∃yφ) ↔ (∃!y∃xφ ∧ ∃yφ)) |
10 | 9 | eubii 1906 | . 2 ⊢ (∃!x∃!y(∃xφ ∧ ∃yφ) ↔ ∃!x(∃!y∃xφ ∧ ∃yφ)) |
11 | ancom 253 | . 2 ⊢ ((∃!x∃yφ ∧ ∃!y∃xφ) ↔ (∃!y∃xφ ∧ ∃!x∃yφ)) | |
12 | 3, 10, 11 | 3bitr4ri 202 | 1 ⊢ ((∃!x∃yφ ∧ ∃!y∃xφ) ↔ ∃!x∃!y(∃xφ ∧ ∃yφ)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 ∃wex 1378 ∃!weu 1897 |
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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