Theorem 2eu4 2287

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 2eu1 2284 for a condition under which the naive definition holds and 2exeu 2281 for a one-way implication. See 2eu5 2288 and 2eu8 2291 for alternate definitions. (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		nfv 1619	. . . 4 ⊢ Ⅎz∃yφ
2	1	eu3 2230	. . 3 ⊢ (∃!x∃yφ ↔ (∃x∃yφ ∧ ∃z∀x(∃yφ → x = z)))
3		nfv 1619	. . . 4 ⊢ Ⅎw∃xφ
4	3	eu3 2230	. . 3 ⊢ (∃!y∃xφ ↔ (∃y∃xφ ∧ ∃w∀y(∃xφ → y = w)))
5	2, 4	anbi12i 678	. 2 ⊢ ((∃!x∃yφ ∧ ∃!y∃xφ) ↔ ((∃x∃yφ ∧ ∃z∀x(∃yφ → x = z)) ∧ (∃y∃xφ ∧ ∃w∀y(∃xφ → y = w))))
6		an4 797	. 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 1741	. . . . 5 ⊢ (∃y∃xφ ↔ ∃x∃yφ)
8	7	anbi2i 675	. . . 4 ⊢ ((∃x∃yφ ∧ ∃y∃xφ) ↔ (∃x∃yφ ∧ ∃x∃yφ))
9		anidm 625	. . . 4 ⊢ ((∃x∃yφ ∧ ∃x∃yφ) ↔ ∃x∃yφ)
10	8, 9	bitri 240	. . 3 ⊢ ((∃x∃yφ ∧ ∃y∃xφ) ↔ ∃x∃yφ)
11		19.26 1593	. . . . . . . 8 ⊢ (∀x(∀y(φ → x = z) ∧ ∀x∀y(φ → y = w)) ↔ (∀x∀y(φ → x = z) ∧ ∀x∀x∀y(φ → y = w)))
12		nfa1 1788	. . . . . . . . . . 11 ⊢ Ⅎx∀x∀y(φ → y = w)
13	12	19.3 1785	. . . . . . . . . 10 ⊢ (∀x∀x∀y(φ → y = w) ↔ ∀x∀y(φ → y = w))
14	13	anbi2i 675	. . . . . . . . 9 ⊢ ((∀x∀y(φ → x = z) ∧ ∀x∀x∀y(φ → y = w)) ↔ (∀x∀y(φ → x = z) ∧ ∀x∀y(φ → y = w)))
15		jcab 833	. . . . . . . . . . . . 13 ⊢ ((φ → (x = z ∧ y = w)) ↔ ((φ → x = z) ∧ (φ → y = w)))
16	15	albii 1566	. . . . . . . . . . . 12 ⊢ (∀y(φ → (x = z ∧ y = w)) ↔ ∀y((φ → x = z) ∧ (φ → y = w)))
17		19.26 1593	. . . . . . . . . . . 12 ⊢ (∀y((φ → x = z) ∧ (φ → y = w)) ↔ (∀y(φ → x = z) ∧ ∀y(φ → y = w)))
18	16, 17	bitri 240	. . . . . . . . . . 11 ⊢ (∀y(φ → (x = z ∧ y = w)) ↔ (∀y(φ → x = z) ∧ ∀y(φ → y = w)))
19	18	albii 1566	. . . . . . . . . 10 ⊢ (∀x∀y(φ → (x = z ∧ y = w)) ↔ ∀x(∀y(φ → x = z) ∧ ∀y(φ → y = w)))
20		19.26 1593	. . . . . . . . . 10 ⊢ (∀x(∀y(φ → x = z) ∧ ∀y(φ → y = w)) ↔ (∀x∀y(φ → x = z) ∧ ∀x∀y(φ → y = w)))
21	19, 20	bitri 240	. . . . . . . . 9 ⊢ (∀x∀y(φ → (x = z ∧ y = w)) ↔ (∀x∀y(φ → x = z) ∧ ∀x∀y(φ → y = w)))
22	14, 21	bitr4i 243	. . . . . . . 8 ⊢ ((∀x∀y(φ → x = z) ∧ ∀x∀x∀y(φ → y = w)) ↔ ∀x∀y(φ → (x = z ∧ y = w)))
23	11, 22	bitr2i 241	. . . . . . 7 ⊢ (∀x∀y(φ → (x = z ∧ y = w)) ↔ ∀x(∀y(φ → x = z) ∧ ∀x∀y(φ → y = w)))
24		19.26 1593	. . . . . . . . 9 ⊢ (∀y(∀y(φ → x = z) ∧ ∀x(φ → y = w)) ↔ (∀y∀y(φ → x = z) ∧ ∀y∀x(φ → y = w)))
25		nfa1 1788	. . . . . . . . . . 11 ⊢ Ⅎy∀y(φ → x = z)
26	25	19.3 1785	. . . . . . . . . 10 ⊢ (∀y∀y(φ → x = z) ↔ ∀y(φ → x = z))
27		alcom 1737	. . . . . . . . . 10 ⊢ (∀y∀x(φ → y = w) ↔ ∀x∀y(φ → y = w))
28	26, 27	anbi12i 678	. . . . . . . . 9 ⊢ ((∀y∀y(φ → x = z) ∧ ∀y∀x(φ → y = w)) ↔ (∀y(φ → x = z) ∧ ∀x∀y(φ → y = w)))
29	24, 28	bitri 240	. . . . . . . 8 ⊢ (∀y(∀y(φ → x = z) ∧ ∀x(φ → y = w)) ↔ (∀y(φ → x = z) ∧ ∀x∀y(φ → y = w)))
30	29	albii 1566	. . . . . . 7 ⊢ (∀x∀y(∀y(φ → x = z) ∧ ∀x(φ → y = w)) ↔ ∀x(∀y(φ → x = z) ∧ ∀x∀y(φ → y = w)))
31	23, 30	bitr4i 243	. . . . . 6 ⊢ (∀x∀y(φ → (x = z ∧ y = w)) ↔ ∀x∀y(∀y(φ → x = z) ∧ ∀x(φ → y = w)))
32		19.23v 1891	. . . . . . . 8 ⊢ (∀y(φ → x = z) ↔ (∃yφ → x = z))
33		19.23v 1891	. . . . . . . 8 ⊢ (∀x(φ → y = w) ↔ (∃xφ → y = w))
34	32, 33	anbi12i 678	. . . . . . 7 ⊢ ((∀y(φ → x = z) ∧ ∀x(φ → y = w)) ↔ ((∃yφ → x = z) ∧ (∃xφ → y = w)))
35	34	2albii 1567	. . . . . 6 ⊢ (∀x∀y(∀y(φ → x = z) ∧ ∀x(φ → y = w)) ↔ ∀x∀y((∃yφ → x = z) ∧ (∃xφ → y = w)))
36		nfe1 1732	. . . . . . . 8 ⊢ Ⅎy∃yφ
37		nfv 1619	. . . . . . . 8 ⊢ Ⅎy x = z
38	36, 37	nfim 1813	. . . . . . 7 ⊢ Ⅎy(∃yφ → x = z)
39		nfe1 1732	. . . . . . . 8 ⊢ Ⅎx∃xφ
40		nfv 1619	. . . . . . . 8 ⊢ Ⅎx y = w
41	39, 40	nfim 1813	. . . . . . 7 ⊢ Ⅎx(∃xφ → y = w)
42	38, 41	aaan 1884	. . . . . 6 ⊢ (∀x∀y((∃yφ → x = z) ∧ (∃xφ → y = w)) ↔ (∀x(∃yφ → x = z) ∧ ∀y(∃xφ → y = w)))
43	31, 35, 42	3bitri 262	. . . . 5 ⊢ (∀x∀y(φ → (x = z ∧ y = w)) ↔ (∀x(∃yφ → x = z) ∧ ∀y(∃xφ → y = w)))
44	43	2exbii 1583	. . . 4 ⊢ (∃z∃w∀x∀y(φ → (x = z ∧ y = w)) ↔ ∃z∃w(∀x(∃yφ → x = z) ∧ ∀y(∃xφ → y = w)))
45		eeanv 1913	. . . 4 ⊢ (∃z∃w(∀x(∃yφ → x = z) ∧ ∀y(∃xφ → y = w)) ↔ (∃z∀x(∃yφ → x = z) ∧ ∃w∀y(∃xφ → y = w)))
46	44, 45	bitr2i 241	. . 3 ⊢ ((∃z∀x(∃yφ → x = z) ∧ ∃w∀y(∃xφ → y = w)) ↔ ∃z∃w∀x∀y(φ → (x = z ∧ y = w)))
47	10, 46	anbi12i 678	. 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))))
48	5, 6, 47	3bitri 262	1 ⊢ ((∃!x∃yφ ∧ ∃!y∃xφ) ↔ (∃x∃yφ ∧ ∃z∃w∀x∀y(φ → (x = z ∧ y = w))))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642  ∃!weu 2204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208

This theorem is referenced by:  2eu5  2288  2eu6  2289