Theorem 2mo 2674

Description: Two ways of expressing "there exists at most one ordered pair ⟨𝑥, 𝑦⟩ such that 𝜑(𝑥, 𝑦) holds. See also 2mo2 2673. (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 2-Nov-2019.)

Assertion

Ref	Expression
2mo	⊢ (∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∀𝑥∀𝑦∀𝑧∀𝑤((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 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 2mo

Step	Hyp	Ref	Expression
1		2mo2 2673	. . . 4 ⊢ ((∃*𝑥∃𝑦𝜑 ∧ ∃*𝑦∃𝑥𝜑) ↔ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
2		nfmo1 2566	. . . . . . 7 ⊢ Ⅎ𝑥∃*𝑥∃𝑦𝜑
3		nftru 1767	. . . . . . . . 9 ⊢ Ⅎ𝑦⊤
4		nfe1 2085	. . . . . . . . . 10 ⊢ Ⅎ𝑥∃𝑥𝜑
5	4	a1i 11	. . . . . . . . 9 ⊢ (⊤ → Ⅎ𝑥∃𝑥𝜑)
6	3, 5	nfmodv 2568	. . . . . . . 8 ⊢ (⊤ → Ⅎ𝑥∃*𝑦∃𝑥𝜑)
7	6	mptru 1514	. . . . . . 7 ⊢ Ⅎ𝑥∃*𝑦∃𝑥𝜑
8	2, 7	nfan 1862	. . . . . 6 ⊢ Ⅎ𝑥(∃*𝑥∃𝑦𝜑 ∧ ∃*𝑦∃𝑥𝜑)
9		nftru 1767	. . . . . . . . . 10 ⊢ Ⅎ𝑥⊤
10		nfe1 2085	. . . . . . . . . . 11 ⊢ Ⅎ𝑦∃𝑦𝜑
11	10	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → Ⅎ𝑦∃𝑦𝜑)
12	9, 11	nfmodv 2568	. . . . . . . . 9 ⊢ (⊤ → Ⅎ𝑦∃*𝑥∃𝑦𝜑)
13	12	mptru 1514	. . . . . . . 8 ⊢ Ⅎ𝑦∃*𝑥∃𝑦𝜑
14		nfmo1 2566	. . . . . . . 8 ⊢ Ⅎ𝑦∃*𝑦∃𝑥𝜑
15	13, 14	nfan 1862	. . . . . . 7 ⊢ Ⅎ𝑦(∃*𝑥∃𝑦𝜑 ∧ ∃*𝑦∃𝑥𝜑)
16		19.8a 2107	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑦𝜑)
17		spsbe 2031	. . . . . . . . . 10 ⊢ ([𝑤 / 𝑦]𝜑 → ∃𝑦𝜑)
18	17	sbimi 2023	. . . . . . . . 9 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 → [𝑧 / 𝑥]∃𝑦𝜑)
19		nfv 1873	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧∃𝑦𝜑
20	19	mo3 2573	. . . . . . . . . . 11 ⊢ (∃*𝑥∃𝑦𝜑 ↔ ∀𝑥∀𝑧((∃𝑦𝜑 ∧ [𝑧 / 𝑥]∃𝑦𝜑) → 𝑥 = 𝑧))
21	20	biimpi 208	. . . . . . . . . 10 ⊢ (∃*𝑥∃𝑦𝜑 → ∀𝑥∀𝑧((∃𝑦𝜑 ∧ [𝑧 / 𝑥]∃𝑦𝜑) → 𝑥 = 𝑧))
22	21	19.21bbi 2116	. . . . . . . . 9 ⊢ (∃*𝑥∃𝑦𝜑 → ((∃𝑦𝜑 ∧ [𝑧 / 𝑥]∃𝑦𝜑) → 𝑥 = 𝑧))
23	16, 18, 22	syl2ani 597	. . . . . . . 8 ⊢ (∃*𝑥∃𝑦𝜑 → ((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → 𝑥 = 𝑧))
24		19.8a 2107	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑥𝜑)
25		sbcom2 2408	. . . . . . . . . 10 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)
26		spsbe 2031	. . . . . . . . . . 11 ⊢ ([𝑧 / 𝑥]𝜑 → ∃𝑥𝜑)
27	26	sbimi 2023	. . . . . . . . . 10 ⊢ ([𝑤 / 𝑦][𝑧 / 𝑥]𝜑 → [𝑤 / 𝑦]∃𝑥𝜑)
28	25, 27	sylbi 209	. . . . . . . . 9 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 → [𝑤 / 𝑦]∃𝑥𝜑)
29		nfv 1873	. . . . . . . . . . . 12 ⊢ Ⅎ𝑤∃𝑥𝜑
30	29	mo3 2573	. . . . . . . . . . 11 ⊢ (∃*𝑦∃𝑥𝜑 ↔ ∀𝑦∀𝑤((∃𝑥𝜑 ∧ [𝑤 / 𝑦]∃𝑥𝜑) → 𝑦 = 𝑤))
31	30	biimpi 208	. . . . . . . . . 10 ⊢ (∃*𝑦∃𝑥𝜑 → ∀𝑦∀𝑤((∃𝑥𝜑 ∧ [𝑤 / 𝑦]∃𝑥𝜑) → 𝑦 = 𝑤))
32	31	19.21bbi 2116	. . . . . . . . 9 ⊢ (∃*𝑦∃𝑥𝜑 → ((∃𝑥𝜑 ∧ [𝑤 / 𝑦]∃𝑥𝜑) → 𝑦 = 𝑤))
33	24, 28, 32	syl2ani 597	. . . . . . . 8 ⊢ (∃*𝑦∃𝑥𝜑 → ((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → 𝑦 = 𝑤))
34	23, 33	anim12ii 608	. . . . . . 7 ⊢ ((∃*𝑥∃𝑦𝜑 ∧ ∃*𝑦∃𝑥𝜑) → ((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
35	15, 34	alrimi 2141	. . . . . 6 ⊢ ((∃*𝑥∃𝑦𝜑 ∧ ∃*𝑦∃𝑥𝜑) → ∀𝑦((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
36	8, 35	alrimi 2141	. . . . 5 ⊢ ((∃*𝑥∃𝑦𝜑 ∧ ∃*𝑦∃𝑥𝜑) → ∀𝑥∀𝑦((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
37	36	alrimivv 1887	. . . 4 ⊢ ((∃*𝑥∃𝑦𝜑 ∧ ∃*𝑦∃𝑥𝜑) → ∀𝑧∀𝑤∀𝑥∀𝑦((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
38	1, 37	sylbir 227	. . 3 ⊢ (∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → ∀𝑧∀𝑤∀𝑥∀𝑦((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
39		nfs1v 2200	. . . . . . . 8 ⊢ Ⅎ𝑥[𝑧 / 𝑥][𝑤 / 𝑦]𝜑
40		nfs1v 2200	. . . . . . . . . 10 ⊢ Ⅎ𝑦[𝑤 / 𝑦]𝜑
41	40	nfsbv 2268	. . . . . . . . 9 ⊢ Ⅎ𝑦[𝑧 / 𝑥][𝑤 / 𝑦]𝜑
42		pm3.21 464	. . . . . . . . . 10 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 → (𝜑 → (𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)))
43	42	imim1d 82	. . . . . . . . 9 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 → (((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
44	41, 43	alimd 2140	. . . . . . . 8 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 → (∀𝑦((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → ∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
45	39, 44	alimd 2140	. . . . . . 7 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 → (∀𝑥∀𝑦((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → ∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
46	45	com12 32	. . . . . 6 ⊢ (∀𝑥∀𝑦((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 → ∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
47	46	aleximi 1794	. . . . 5 ⊢ (∀𝑤∀𝑥∀𝑦((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → (∃𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑 → ∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
48	47	aleximi 1794	. . . 4 ⊢ (∀𝑧∀𝑤∀𝑥∀𝑦((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → (∃𝑧∃𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑 → ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
49		2nexaln 1792	. . . . . 6 ⊢ (¬ ∃𝑥∃𝑦𝜑 ↔ ∀𝑥∀𝑦 ¬ 𝜑)
50		nfv 1873	. . . . . . 7 ⊢ Ⅎ𝑤𝜑
51		nfv 1873	. . . . . . 7 ⊢ Ⅎ𝑧𝜑
52	50, 51	2sb8ev 2286	. . . . . 6 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
53	49, 52	xchnxbi 324	. . . . 5 ⊢ (¬ ∃𝑧∃𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∀𝑥∀𝑦 ¬ 𝜑)
54		pm2.21 121	. . . . . . . . 9 ⊢ (¬ 𝜑 → (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
55	54	2alimi 1775	. . . . . . . 8 ⊢ (∀𝑥∀𝑦 ¬ 𝜑 → ∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
56	55	2eximi 1798	. . . . . . 7 ⊢ (∃𝑧∃𝑤∀𝑥∀𝑦 ¬ 𝜑 → ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
57	56	19.23bi 2117	. . . . . 6 ⊢ (∃𝑤∀𝑥∀𝑦 ¬ 𝜑 → ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
58	57	19.23bi 2117	. . . . 5 ⊢ (∀𝑥∀𝑦 ¬ 𝜑 → ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
59	53, 58	sylbi 209	. . . 4 ⊢ (¬ ∃𝑧∃𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑 → ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
60	48, 59	pm2.61d1 173	. . 3 ⊢ (∀𝑧∀𝑤∀𝑥∀𝑦((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
61	38, 60	impbii 201	. 2 ⊢ (∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∀𝑧∀𝑤∀𝑥∀𝑦((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
62		alrot4 2095	. 2 ⊢ (∀𝑧∀𝑤∀𝑥∀𝑦((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∀𝑥∀𝑦∀𝑧∀𝑤((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
63	61, 62	bitri 267	1 ⊢ (∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∀𝑥∀𝑦∀𝑧∀𝑤((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 387  ∀wal 1505  ⊤wtru 1508  ∃wex 1742  Ⅎwnf 1746  [wsb 2013  ∃*wmo 2542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-10 2077  ax-11 2091  ax-12 2104

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544

This theorem is referenced by:  2mos  2675