Theorem 2eu4 1070

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 1067 for a condition under which the naive definition holds and 2exeu 1066 for a one-way implication. See 2eu5 1071 for an alternate definition.

Assertion

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

Distinct variable group(s):   x,y,z,w   φ,z,w

Proof of Theorem 2eu4

Step	Hyp	Ref	Expression
1		ax-17 925	. . . 4 ⊢ (∃yφ → ∀z∃yφ)
2	1	eu3 1024	. . 3 ⊢ (∃!x∃yφ ↔ (∃x∃yφ ∧ ∃z∀x(∃yφ → x = z)))
3		ax-17 925	. . . 4 ⊢ (∃xφ → ∀w∃xφ)
4	3	eu3 1024	. . 3 ⊢ (∃!y∃xφ ↔ (∃y∃xφ ∧ ∃w∀y(∃xφ → y = w)))
5	2, 4	anbi12i 369	. 2 ⊢ ((∃!x∃yφ ∧ ∃!y∃xφ) ↔ ((∃x∃yφ ∧ ∃z∀x(∃yφ → x = z)) ∧ (∃y∃xφ ∧ ∃w∀y(∃xφ → y = w))))
6		an4 388	. 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 728	. . . . 5 ⊢ (∃y∃xφ ↔ ∃x∃yφ)
8	7	anbi2i 367	. . . 4 ⊢ ((∃x∃yφ ∧ ∃y∃xφ) ↔ (∃x∃yφ ∧ ∃x∃yφ))
9		anidm 331	. . . 4 ⊢ ((∃x∃yφ ∧ ∃x∃yφ) ↔ ∃x∃yφ)
10	8, 9	bitr 151	. . 3 ⊢ ((∃x∃yφ ∧ ∃y∃xφ) ↔ ∃x∃yφ)
11		hba1 698	. . . . . . . . . 10 ⊢ (∀x∀y(φ → y = w) → ∀x∀x∀y(φ → y = w))
12	11	19.3r 714	. . . . . . . . 9 ⊢ (∀x∀y(φ → y = w) ↔ ∀x∀x∀y(φ → y = w))
13	12	anbi2i 367	. . . . . . . 8 ⊢ ((∀x∀y(φ → x = z) ∧ ∀x∀y(φ → y = w)) ↔ (∀x∀y(φ → x = z) ∧ ∀x∀x∀y(φ → y = w)))
14		jcab 454	. . . . . . . . . . . 12 ⊢ ((φ → (x = z ∧ y = w)) ↔ ((φ → x = z) ∧ (φ → y = w)))
15	14	bial 695	. . . . . . . . . . 11 ⊢ (∀y(φ → (x = z ∧ y = w)) ↔ ∀y((φ → x = z) ∧ (φ → y = w)))
16		19.26 749	. . . . . . . . . . 11 ⊢ (∀y((φ → x = z) ∧ (φ → y = w)) ↔ (∀y(φ → x = z) ∧ ∀y(φ → y = w)))
17	15, 16	bitr 151	. . . . . . . . . 10 ⊢ (∀y(φ → (x = z ∧ y = w)) ↔ (∀y(φ → x = z) ∧ ∀y(φ → y = w)))
18	17	bial 695	. . . . . . . . 9 ⊢ (∀x∀y(φ → (x = z ∧ y = w)) ↔ ∀x(∀y(φ → x = z) ∧ ∀y(φ → y = w)))
19		19.26 749	. . . . . . . . 9 ⊢ (∀x(∀y(φ → x = z) ∧ ∀y(φ → y = w)) ↔ (∀x∀y(φ → x = z) ∧ ∀x∀y(φ → y = w)))
20	18, 19	bitr 151	. . . . . . . 8 ⊢ (∀x∀y(φ → (x = z ∧ y = w)) ↔ (∀x∀y(φ → x = z) ∧ ∀x∀y(φ → y = w)))
21		19.26 749	. . . . . . . 8 ⊢ (∀x(∀y(φ → x = z) ∧ ∀x∀y(φ → y = w)) ↔ (∀x∀y(φ → x = z) ∧ ∀x∀x∀y(φ → y = w)))
22	13, 20, 21	3bitr4 158	. . . . . . 7 ⊢ (∀x∀y(φ → (x = z ∧ y = w)) ↔ ∀x(∀y(φ → x = z) ∧ ∀x∀y(φ → y = w)))
23		19.26 749	. . . . . . . . 9 ⊢ (∀y(∀y(φ → x = z) ∧ ∀x(φ → y = w)) ↔ (∀y∀y(φ → x = z) ∧ ∀y∀x(φ → y = w)))
24		ax-4 673	. . . . . . . . . . 11 ⊢ (∀y∀y(φ → x = z) → ∀y(φ → x = z))
25		hba1 698	. . . . . . . . . . 11 ⊢ (∀y(φ → x = z) → ∀y∀y(φ → x = z))
26	24, 25	impbi 139	. . . . . . . . . 10 ⊢ (∀y∀y(φ → x = z) ↔ ∀y(φ → x = z))
27		alcom 715	. . . . . . . . . 10 ⊢ (∀y∀x(φ → y = w) ↔ ∀x∀y(φ → y = w))
28	26, 27	anbi12i 369	. . . . . . . . 9 ⊢ ((∀y∀y(φ → x = z) ∧ ∀y∀x(φ → y = w)) ↔ (∀y(φ → x = z) ∧ ∀x∀y(φ → y = w)))
29	23, 28	bitr 151	. . . . . . . 8 ⊢ (∀y(∀y(φ → x = z) ∧ ∀x(φ → y = w)) ↔ (∀y(φ → x = z) ∧ ∀x∀y(φ → y = w)))
30	29	bial 695	. . . . . . 7 ⊢ (∀x∀y(∀y(φ → x = z) ∧ ∀x(φ → y = w)) ↔ ∀x(∀y(φ → x = z) ∧ ∀x∀y(φ → y = w)))
31	22, 30	bitr4 154	. . . . . 6 ⊢ (∀x∀y(φ → (x = z ∧ y = w)) ↔ ∀x∀y(∀y(φ → x = z) ∧ ∀x(φ → y = w)))
32		19.23v 950	. . . . . . . 8 ⊢ (∀y(φ → x = z) ↔ (∃yφ → x = z))
33		19.23v 950	. . . . . . . 8 ⊢ (∀x(φ → y = w) ↔ (∃xφ → y = w))
34	32, 33	anbi12i 369	. . . . . . 7 ⊢ ((∀y(φ → x = z) ∧ ∀x(φ → y = w)) ↔ ((∃yφ → x = z) ∧ (∃xφ → y = w)))
35	34	bi2al 696	. . . . . 6 ⊢ (∀x∀y(∀y(φ → x = z) ∧ ∀x(φ → y = w)) ↔ ∀x∀y((∃yφ → x = z) ∧ (∃xφ → y = w)))
36		hbe1 709	. . . . . . . 8 ⊢ (∃yφ → ∀y∃yφ)
37		ax-17 925	. . . . . . . 8 ⊢ (x = z → ∀y x = z)
38	36, 37	hbim 702	. . . . . . 7 ⊢ ((∃yφ → x = z) → ∀y(∃yφ → x = z))
39		hbe1 709	. . . . . . . 8 ⊢ (∃xφ → ∀x∃xφ)
40		ax-17 925	. . . . . . . 8 ⊢ (y = w → ∀x y = w)
41	39, 40	hbim 702	. . . . . . 7 ⊢ ((∃xφ → y = w) → ∀x(∃xφ → y = w))
42	38, 41	aaan 794	. . . . . 6 ⊢ (∀x∀y((∃yφ → x = z) ∧ (∃xφ → y = w)) ↔ (∀x(∃yφ → x = z) ∧ ∀y(∃xφ → y = w)))
43	31, 35, 42	3bitr 155	. . . . 5 ⊢ (∀x∀y(φ → (x = z ∧ y = w)) ↔ (∀x(∃yφ → x = z) ∧ ∀y(∃xφ → y = w)))
44	43	bi2ex 734	. . . 4 ⊢ (∃z∃w∀x∀y(φ → (x = z ∧ y = w)) ↔ ∃z∃w(∀x(∃yφ → x = z) ∧ ∀y(∃xφ → y = w)))
45		eeanv 980	. . . 4 ⊢ (∃z∃w(∀x(∃yφ → x = z) ∧ ∀y(∃xφ → y = w)) ↔ (∃z∀x(∃yφ → x = z) ∧ ∃w∀y(∃xφ → y = w)))
46	44, 45	bitr2 152	. . 3 ⊢ ((∃z∀x(∃yφ → x = z) ∧ ∃w∀y(∃xφ → y = w)) ↔ ∃z∃w∀x∀y(φ → (x = z ∧ y = w)))
47	10, 46	anbi12i 369	. 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	3bitr 155	1 ⊢ ((∃!x∃yφ ∧ ∃!y∃xφ) ↔ (∃x∃yφ ∧ ∃z∃w∀x∀y(φ → (x = z ∧ y = w))))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797  ∃!weu 1007

This theorem is referenced by:  2eu5 1071

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009

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