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Theorem 2eu4 2009
Description: This theorem provides us with a definition of double existential uniqueness ("exactly one 𝑥 and exactly one 𝑦"). Naively one might think (incorrectly) that it could be defined by ∃!𝑥∃!𝑦𝜑. See 2exeu 2008 for a one-way implication. (Contributed by NM, 3-Dec-2001.)
Assertion
Ref Expression
2eu4 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) ↔ (∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑤   𝜑,𝑧,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 2eu4
StepHypRef Expression
1 ax-17 1435 . . . 4 (∃𝑦𝜑 → ∀𝑧𝑦𝜑)
21eu3h 1961 . . 3 (∃!𝑥𝑦𝜑 ↔ (∃𝑥𝑦𝜑 ∧ ∃𝑧𝑥(∃𝑦𝜑𝑥 = 𝑧)))
3 ax-17 1435 . . . 4 (∃𝑥𝜑 → ∀𝑤𝑥𝜑)
43eu3h 1961 . . 3 (∃!𝑦𝑥𝜑 ↔ (∃𝑦𝑥𝜑 ∧ ∃𝑤𝑦(∃𝑥𝜑𝑦 = 𝑤)))
52, 4anbi12i 441 . 2 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) ↔ ((∃𝑥𝑦𝜑 ∧ ∃𝑧𝑥(∃𝑦𝜑𝑥 = 𝑧)) ∧ (∃𝑦𝑥𝜑 ∧ ∃𝑤𝑦(∃𝑥𝜑𝑦 = 𝑤))))
6 an4 528 . 2 (((∃𝑥𝑦𝜑 ∧ ∃𝑧𝑥(∃𝑦𝜑𝑥 = 𝑧)) ∧ (∃𝑦𝑥𝜑 ∧ ∃𝑤𝑦(∃𝑥𝜑𝑦 = 𝑤))) ↔ ((∃𝑥𝑦𝜑 ∧ ∃𝑦𝑥𝜑) ∧ (∃𝑧𝑥(∃𝑦𝜑𝑥 = 𝑧) ∧ ∃𝑤𝑦(∃𝑥𝜑𝑦 = 𝑤))))
7 excom 1570 . . . . 5 (∃𝑦𝑥𝜑 ↔ ∃𝑥𝑦𝜑)
87anbi2i 438 . . . 4 ((∃𝑥𝑦𝜑 ∧ ∃𝑦𝑥𝜑) ↔ (∃𝑥𝑦𝜑 ∧ ∃𝑥𝑦𝜑))
9 anidm 382 . . . 4 ((∃𝑥𝑦𝜑 ∧ ∃𝑥𝑦𝜑) ↔ ∃𝑥𝑦𝜑)
108, 9bitri 177 . . 3 ((∃𝑥𝑦𝜑 ∧ ∃𝑦𝑥𝜑) ↔ ∃𝑥𝑦𝜑)
11 hba1 1449 . . . . . . . . . 10 (∀𝑥𝑦(𝜑𝑦 = 𝑤) → ∀𝑥𝑥𝑦(𝜑𝑦 = 𝑤))
121119.3h 1461 . . . . . . . . 9 (∀𝑥𝑥𝑦(𝜑𝑦 = 𝑤) ↔ ∀𝑥𝑦(𝜑𝑦 = 𝑤))
1312anbi2i 438 . . . . . . . 8 ((∀𝑥𝑦(𝜑𝑥 = 𝑧) ∧ ∀𝑥𝑥𝑦(𝜑𝑦 = 𝑤)) ↔ (∀𝑥𝑦(𝜑𝑥 = 𝑧) ∧ ∀𝑥𝑦(𝜑𝑦 = 𝑤)))
14 19.26 1386 . . . . . . . 8 (∀𝑥(∀𝑦(𝜑𝑥 = 𝑧) ∧ ∀𝑥𝑦(𝜑𝑦 = 𝑤)) ↔ (∀𝑥𝑦(𝜑𝑥 = 𝑧) ∧ ∀𝑥𝑥𝑦(𝜑𝑦 = 𝑤)))
15 jcab 545 . . . . . . . . . . . 12 ((𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ((𝜑𝑥 = 𝑧) ∧ (𝜑𝑦 = 𝑤)))
1615albii 1375 . . . . . . . . . . 11 (∀𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∀𝑦((𝜑𝑥 = 𝑧) ∧ (𝜑𝑦 = 𝑤)))
17 19.26 1386 . . . . . . . . . . 11 (∀𝑦((𝜑𝑥 = 𝑧) ∧ (𝜑𝑦 = 𝑤)) ↔ (∀𝑦(𝜑𝑥 = 𝑧) ∧ ∀𝑦(𝜑𝑦 = 𝑤)))
1816, 17bitri 177 . . . . . . . . . 10 (∀𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ (∀𝑦(𝜑𝑥 = 𝑧) ∧ ∀𝑦(𝜑𝑦 = 𝑤)))
1918albii 1375 . . . . . . . . 9 (∀𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∀𝑥(∀𝑦(𝜑𝑥 = 𝑧) ∧ ∀𝑦(𝜑𝑦 = 𝑤)))
20 19.26 1386 . . . . . . . . 9 (∀𝑥(∀𝑦(𝜑𝑥 = 𝑧) ∧ ∀𝑦(𝜑𝑦 = 𝑤)) ↔ (∀𝑥𝑦(𝜑𝑥 = 𝑧) ∧ ∀𝑥𝑦(𝜑𝑦 = 𝑤)))
2119, 20bitri 177 . . . . . . . 8 (∀𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ (∀𝑥𝑦(𝜑𝑥 = 𝑧) ∧ ∀𝑥𝑦(𝜑𝑦 = 𝑤)))
2213, 14, 213bitr4ri 206 . . . . . . 7 (∀𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∀𝑥(∀𝑦(𝜑𝑥 = 𝑧) ∧ ∀𝑥𝑦(𝜑𝑦 = 𝑤)))
23 19.26 1386 . . . . . . . . 9 (∀𝑦(∀𝑦(𝜑𝑥 = 𝑧) ∧ ∀𝑥(𝜑𝑦 = 𝑤)) ↔ (∀𝑦𝑦(𝜑𝑥 = 𝑧) ∧ ∀𝑦𝑥(𝜑𝑦 = 𝑤)))
24 hba1 1449 . . . . . . . . . . 11 (∀𝑦(𝜑𝑥 = 𝑧) → ∀𝑦𝑦(𝜑𝑥 = 𝑧))
252419.3h 1461 . . . . . . . . . 10 (∀𝑦𝑦(𝜑𝑥 = 𝑧) ↔ ∀𝑦(𝜑𝑥 = 𝑧))
26 alcom 1383 . . . . . . . . . 10 (∀𝑦𝑥(𝜑𝑦 = 𝑤) ↔ ∀𝑥𝑦(𝜑𝑦 = 𝑤))
2725, 26anbi12i 441 . . . . . . . . 9 ((∀𝑦𝑦(𝜑𝑥 = 𝑧) ∧ ∀𝑦𝑥(𝜑𝑦 = 𝑤)) ↔ (∀𝑦(𝜑𝑥 = 𝑧) ∧ ∀𝑥𝑦(𝜑𝑦 = 𝑤)))
2823, 27bitri 177 . . . . . . . 8 (∀𝑦(∀𝑦(𝜑𝑥 = 𝑧) ∧ ∀𝑥(𝜑𝑦 = 𝑤)) ↔ (∀𝑦(𝜑𝑥 = 𝑧) ∧ ∀𝑥𝑦(𝜑𝑦 = 𝑤)))
2928albii 1375 . . . . . . 7 (∀𝑥𝑦(∀𝑦(𝜑𝑥 = 𝑧) ∧ ∀𝑥(𝜑𝑦 = 𝑤)) ↔ ∀𝑥(∀𝑦(𝜑𝑥 = 𝑧) ∧ ∀𝑥𝑦(𝜑𝑦 = 𝑤)))
3022, 29bitr4i 180 . . . . . 6 (∀𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∀𝑥𝑦(∀𝑦(𝜑𝑥 = 𝑧) ∧ ∀𝑥(𝜑𝑦 = 𝑤)))
31 19.23v 1779 . . . . . . . 8 (∀𝑦(𝜑𝑥 = 𝑧) ↔ (∃𝑦𝜑𝑥 = 𝑧))
32 19.23v 1779 . . . . . . . 8 (∀𝑥(𝜑𝑦 = 𝑤) ↔ (∃𝑥𝜑𝑦 = 𝑤))
3331, 32anbi12i 441 . . . . . . 7 ((∀𝑦(𝜑𝑥 = 𝑧) ∧ ∀𝑥(𝜑𝑦 = 𝑤)) ↔ ((∃𝑦𝜑𝑥 = 𝑧) ∧ (∃𝑥𝜑𝑦 = 𝑤)))
34332albii 1376 . . . . . 6 (∀𝑥𝑦(∀𝑦(𝜑𝑥 = 𝑧) ∧ ∀𝑥(𝜑𝑦 = 𝑤)) ↔ ∀𝑥𝑦((∃𝑦𝜑𝑥 = 𝑧) ∧ (∃𝑥𝜑𝑦 = 𝑤)))
35 hbe1 1400 . . . . . . . 8 (∃𝑦𝜑 → ∀𝑦𝑦𝜑)
36 ax-17 1435 . . . . . . . 8 (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧)
3735, 36hbim 1453 . . . . . . 7 ((∃𝑦𝜑𝑥 = 𝑧) → ∀𝑦(∃𝑦𝜑𝑥 = 𝑧))
38 hbe1 1400 . . . . . . . 8 (∃𝑥𝜑 → ∀𝑥𝑥𝜑)
39 ax-17 1435 . . . . . . . 8 (𝑦 = 𝑤 → ∀𝑥 𝑦 = 𝑤)
4038, 39hbim 1453 . . . . . . 7 ((∃𝑥𝜑𝑦 = 𝑤) → ∀𝑥(∃𝑥𝜑𝑦 = 𝑤))
4137, 40aaanh 1494 . . . . . 6 (∀𝑥𝑦((∃𝑦𝜑𝑥 = 𝑧) ∧ (∃𝑥𝜑𝑦 = 𝑤)) ↔ (∀𝑥(∃𝑦𝜑𝑥 = 𝑧) ∧ ∀𝑦(∃𝑥𝜑𝑦 = 𝑤)))
4230, 34, 413bitri 199 . . . . 5 (∀𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ (∀𝑥(∃𝑦𝜑𝑥 = 𝑧) ∧ ∀𝑦(∃𝑥𝜑𝑦 = 𝑤)))
43422exbii 1513 . . . 4 (∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∃𝑧𝑤(∀𝑥(∃𝑦𝜑𝑥 = 𝑧) ∧ ∀𝑦(∃𝑥𝜑𝑦 = 𝑤)))
44 eeanv 1823 . . . 4 (∃𝑧𝑤(∀𝑥(∃𝑦𝜑𝑥 = 𝑧) ∧ ∀𝑦(∃𝑥𝜑𝑦 = 𝑤)) ↔ (∃𝑧𝑥(∃𝑦𝜑𝑥 = 𝑧) ∧ ∃𝑤𝑦(∃𝑥𝜑𝑦 = 𝑤)))
4543, 44bitr2i 178 . . 3 ((∃𝑧𝑥(∃𝑦𝜑𝑥 = 𝑧) ∧ ∃𝑤𝑦(∃𝑥𝜑𝑦 = 𝑤)) ↔ ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)))
4610, 45anbi12i 441 . 2 (((∃𝑥𝑦𝜑 ∧ ∃𝑦𝑥𝜑) ∧ (∃𝑧𝑥(∃𝑦𝜑𝑥 = 𝑧) ∧ ∃𝑤𝑦(∃𝑥𝜑𝑦 = 𝑤))) ↔ (∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))
475, 6, 463bitri 199 1 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) ↔ (∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1257  wex 1397  ∃!weu 1916
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-eu 1919
This theorem is referenced by: (None)
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