Theorem 2eu4 2112

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 2111 for a one-way implication. (Contributed by NM, 3-Dec-2001.)

Assertion

Ref	Expression
2eu4	⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))

Distinct variable groups:   𝑥,𝑦,𝑧,𝑤   𝜑,𝑧,𝑤

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 2eu4

Step	Hyp	Ref	Expression
1		ax-17 1519	. . . 4 ⊢ (∃𝑦𝜑 → ∀𝑧∃𝑦𝜑)
2	1	eu3h 2064	. . 3 ⊢ (∃!𝑥∃𝑦𝜑 ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∀𝑥(∃𝑦𝜑 → 𝑥 = 𝑧)))
3		ax-17 1519	. . . 4 ⊢ (∃𝑥𝜑 → ∀𝑤∃𝑥𝜑)
4	3	eu3h 2064	. . 3 ⊢ (∃!𝑦∃𝑥𝜑 ↔ (∃𝑦∃𝑥𝜑 ∧ ∃𝑤∀𝑦(∃𝑥𝜑 → 𝑦 = 𝑤)))
5	2, 4	anbi12i 457	. 2 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ ((∃𝑥∃𝑦𝜑 ∧ ∃𝑧∀𝑥(∃𝑦𝜑 → 𝑥 = 𝑧)) ∧ (∃𝑦∃𝑥𝜑 ∧ ∃𝑤∀𝑦(∃𝑥𝜑 → 𝑦 = 𝑤))))
6		an4 581	. 2 ⊢ (((∃𝑥∃𝑦𝜑 ∧ ∃𝑧∀𝑥(∃𝑦𝜑 → 𝑥 = 𝑧)) ∧ (∃𝑦∃𝑥𝜑 ∧ ∃𝑤∀𝑦(∃𝑥𝜑 → 𝑦 = 𝑤))) ↔ ((∃𝑥∃𝑦𝜑 ∧ ∃𝑦∃𝑥𝜑) ∧ (∃𝑧∀𝑥(∃𝑦𝜑 → 𝑥 = 𝑧) ∧ ∃𝑤∀𝑦(∃𝑥𝜑 → 𝑦 = 𝑤))))
7		excom 1657	. . . . 5 ⊢ (∃𝑦∃𝑥𝜑 ↔ ∃𝑥∃𝑦𝜑)
8	7	anbi2i 454	. . . 4 ⊢ ((∃𝑥∃𝑦𝜑 ∧ ∃𝑦∃𝑥𝜑) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑥∃𝑦𝜑))
9		anidm 394	. . . 4 ⊢ ((∃𝑥∃𝑦𝜑 ∧ ∃𝑥∃𝑦𝜑) ↔ ∃𝑥∃𝑦𝜑)
10	8, 9	bitri 183	. . 3 ⊢ ((∃𝑥∃𝑦𝜑 ∧ ∃𝑦∃𝑥𝜑) ↔ ∃𝑥∃𝑦𝜑)
11		hba1 1533	. . . . . . . . . 10 ⊢ (∀𝑥∀𝑦(𝜑 → 𝑦 = 𝑤) → ∀𝑥∀𝑥∀𝑦(𝜑 → 𝑦 = 𝑤))
12	11	19.3h 1546	. . . . . . . . 9 ⊢ (∀𝑥∀𝑥∀𝑦(𝜑 → 𝑦 = 𝑤) ↔ ∀𝑥∀𝑦(𝜑 → 𝑦 = 𝑤))
13	12	anbi2i 454	. . . . . . . 8 ⊢ ((∀𝑥∀𝑦(𝜑 → 𝑥 = 𝑧) ∧ ∀𝑥∀𝑥∀𝑦(𝜑 → 𝑦 = 𝑤)) ↔ (∀𝑥∀𝑦(𝜑 → 𝑥 = 𝑧) ∧ ∀𝑥∀𝑦(𝜑 → 𝑦 = 𝑤)))
14		19.26 1474	. . . . . . . 8 ⊢ (∀𝑥(∀𝑦(𝜑 → 𝑥 = 𝑧) ∧ ∀𝑥∀𝑦(𝜑 → 𝑦 = 𝑤)) ↔ (∀𝑥∀𝑦(𝜑 → 𝑥 = 𝑧) ∧ ∀𝑥∀𝑥∀𝑦(𝜑 → 𝑦 = 𝑤)))
15		jcab 598	. . . . . . . . . . . 12 ⊢ ((𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ((𝜑 → 𝑥 = 𝑧) ∧ (𝜑 → 𝑦 = 𝑤)))
16	15	albii 1463	. . . . . . . . . . 11 ⊢ (∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∀𝑦((𝜑 → 𝑥 = 𝑧) ∧ (𝜑 → 𝑦 = 𝑤)))
17		19.26 1474	. . . . . . . . . . 11 ⊢ (∀𝑦((𝜑 → 𝑥 = 𝑧) ∧ (𝜑 → 𝑦 = 𝑤)) ↔ (∀𝑦(𝜑 → 𝑥 = 𝑧) ∧ ∀𝑦(𝜑 → 𝑦 = 𝑤)))
18	16, 17	bitri 183	. . . . . . . . . 10 ⊢ (∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ (∀𝑦(𝜑 → 𝑥 = 𝑧) ∧ ∀𝑦(𝜑 → 𝑦 = 𝑤)))
19	18	albii 1463	. . . . . . . . 9 ⊢ (∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∀𝑥(∀𝑦(𝜑 → 𝑥 = 𝑧) ∧ ∀𝑦(𝜑 → 𝑦 = 𝑤)))
20		19.26 1474	. . . . . . . . 9 ⊢ (∀𝑥(∀𝑦(𝜑 → 𝑥 = 𝑧) ∧ ∀𝑦(𝜑 → 𝑦 = 𝑤)) ↔ (∀𝑥∀𝑦(𝜑 → 𝑥 = 𝑧) ∧ ∀𝑥∀𝑦(𝜑 → 𝑦 = 𝑤)))
21	19, 20	bitri 183	. . . . . . . 8 ⊢ (∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ (∀𝑥∀𝑦(𝜑 → 𝑥 = 𝑧) ∧ ∀𝑥∀𝑦(𝜑 → 𝑦 = 𝑤)))
22	13, 14, 21	3bitr4ri 212	. . . . . . 7 ⊢ (∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∀𝑥(∀𝑦(𝜑 → 𝑥 = 𝑧) ∧ ∀𝑥∀𝑦(𝜑 → 𝑦 = 𝑤)))
23		19.26 1474	. . . . . . . . 9 ⊢ (∀𝑦(∀𝑦(𝜑 → 𝑥 = 𝑧) ∧ ∀𝑥(𝜑 → 𝑦 = 𝑤)) ↔ (∀𝑦∀𝑦(𝜑 → 𝑥 = 𝑧) ∧ ∀𝑦∀𝑥(𝜑 → 𝑦 = 𝑤)))
24		hba1 1533	. . . . . . . . . . 11 ⊢ (∀𝑦(𝜑 → 𝑥 = 𝑧) → ∀𝑦∀𝑦(𝜑 → 𝑥 = 𝑧))
25	24	19.3h 1546	. . . . . . . . . 10 ⊢ (∀𝑦∀𝑦(𝜑 → 𝑥 = 𝑧) ↔ ∀𝑦(𝜑 → 𝑥 = 𝑧))
26		alcom 1471	. . . . . . . . . 10 ⊢ (∀𝑦∀𝑥(𝜑 → 𝑦 = 𝑤) ↔ ∀𝑥∀𝑦(𝜑 → 𝑦 = 𝑤))
27	25, 26	anbi12i 457	. . . . . . . . 9 ⊢ ((∀𝑦∀𝑦(𝜑 → 𝑥 = 𝑧) ∧ ∀𝑦∀𝑥(𝜑 → 𝑦 = 𝑤)) ↔ (∀𝑦(𝜑 → 𝑥 = 𝑧) ∧ ∀𝑥∀𝑦(𝜑 → 𝑦 = 𝑤)))
28	23, 27	bitri 183	. . . . . . . 8 ⊢ (∀𝑦(∀𝑦(𝜑 → 𝑥 = 𝑧) ∧ ∀𝑥(𝜑 → 𝑦 = 𝑤)) ↔ (∀𝑦(𝜑 → 𝑥 = 𝑧) ∧ ∀𝑥∀𝑦(𝜑 → 𝑦 = 𝑤)))
29	28	albii 1463	. . . . . . 7 ⊢ (∀𝑥∀𝑦(∀𝑦(𝜑 → 𝑥 = 𝑧) ∧ ∀𝑥(𝜑 → 𝑦 = 𝑤)) ↔ ∀𝑥(∀𝑦(𝜑 → 𝑥 = 𝑧) ∧ ∀𝑥∀𝑦(𝜑 → 𝑦 = 𝑤)))
30	22, 29	bitr4i 186	. . . . . 6 ⊢ (∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∀𝑥∀𝑦(∀𝑦(𝜑 → 𝑥 = 𝑧) ∧ ∀𝑥(𝜑 → 𝑦 = 𝑤)))
31		19.23v 1876	. . . . . . . 8 ⊢ (∀𝑦(𝜑 → 𝑥 = 𝑧) ↔ (∃𝑦𝜑 → 𝑥 = 𝑧))
32		19.23v 1876	. . . . . . . 8 ⊢ (∀𝑥(𝜑 → 𝑦 = 𝑤) ↔ (∃𝑥𝜑 → 𝑦 = 𝑤))
33	31, 32	anbi12i 457	. . . . . . 7 ⊢ ((∀𝑦(𝜑 → 𝑥 = 𝑧) ∧ ∀𝑥(𝜑 → 𝑦 = 𝑤)) ↔ ((∃𝑦𝜑 → 𝑥 = 𝑧) ∧ (∃𝑥𝜑 → 𝑦 = 𝑤)))
34	33	2albii 1464	. . . . . 6 ⊢ (∀𝑥∀𝑦(∀𝑦(𝜑 → 𝑥 = 𝑧) ∧ ∀𝑥(𝜑 → 𝑦 = 𝑤)) ↔ ∀𝑥∀𝑦((∃𝑦𝜑 → 𝑥 = 𝑧) ∧ (∃𝑥𝜑 → 𝑦 = 𝑤)))
35		hbe1 1488	. . . . . . . 8 ⊢ (∃𝑦𝜑 → ∀𝑦∃𝑦𝜑)
36		ax-17 1519	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧)
37	35, 36	hbim 1538	. . . . . . 7 ⊢ ((∃𝑦𝜑 → 𝑥 = 𝑧) → ∀𝑦(∃𝑦𝜑 → 𝑥 = 𝑧))
38		hbe1 1488	. . . . . . . 8 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)
39		ax-17 1519	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → ∀𝑥 𝑦 = 𝑤)
40	38, 39	hbim 1538	. . . . . . 7 ⊢ ((∃𝑥𝜑 → 𝑦 = 𝑤) → ∀𝑥(∃𝑥𝜑 → 𝑦 = 𝑤))
41	37, 40	aaanh 1579	. . . . . 6 ⊢ (∀𝑥∀𝑦((∃𝑦𝜑 → 𝑥 = 𝑧) ∧ (∃𝑥𝜑 → 𝑦 = 𝑤)) ↔ (∀𝑥(∃𝑦𝜑 → 𝑥 = 𝑧) ∧ ∀𝑦(∃𝑥𝜑 → 𝑦 = 𝑤)))
42	30, 34, 41	3bitri 205	. . . . 5 ⊢ (∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ (∀𝑥(∃𝑦𝜑 → 𝑥 = 𝑧) ∧ ∀𝑦(∃𝑥𝜑 → 𝑦 = 𝑤)))
43	42	2exbii 1599	. . . 4 ⊢ (∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑧∃𝑤(∀𝑥(∃𝑦𝜑 → 𝑥 = 𝑧) ∧ ∀𝑦(∃𝑥𝜑 → 𝑦 = 𝑤)))
44		eeanv 1925	. . . 4 ⊢ (∃𝑧∃𝑤(∀𝑥(∃𝑦𝜑 → 𝑥 = 𝑧) ∧ ∀𝑦(∃𝑥𝜑 → 𝑦 = 𝑤)) ↔ (∃𝑧∀𝑥(∃𝑦𝜑 → 𝑥 = 𝑧) ∧ ∃𝑤∀𝑦(∃𝑥𝜑 → 𝑦 = 𝑤)))
45	43, 44	bitr2i 184	. . 3 ⊢ ((∃𝑧∀𝑥(∃𝑦𝜑 → 𝑥 = 𝑧) ∧ ∃𝑤∀𝑦(∃𝑥𝜑 → 𝑦 = 𝑤)) ↔ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
46	10, 45	anbi12i 457	. 2 ⊢ (((∃𝑥∃𝑦𝜑 ∧ ∃𝑦∃𝑥𝜑) ∧ (∃𝑧∀𝑥(∃𝑦𝜑 → 𝑥 = 𝑧) ∧ ∃𝑤∀𝑦(∃𝑥𝜑 → 𝑦 = 𝑤))) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
47	5, 6, 46	3bitri 205	1 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1346  ∃wex 1485  ∃!weu 2019

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-eu 2022

This theorem is referenced by: (None)