Theorem copsexg 4062

Description: Substitution of class 𝐴 for ordered pair ⟨𝑥, 𝑦⟩. (Contributed by NM, 27-Dec-1996.) (Revised by Andrew Salmon, 11-Jul-2011.)

Assertion

Ref	Expression
copsexg	⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))

Distinct variable groups:   𝑥,𝐴   𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem copsexg

Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2622	. . . 4 ⊢ 𝑥 ∈ V
2		vex 2622	. . . 4 ⊢ 𝑦 ∈ V
3	1, 2	eqvinop 4061	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑧∃𝑤(𝐴 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩))
4		19.8a 1527	. . . . . . . . 9 ⊢ (∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
5	4	19.23bi 1528	. . . . . . . 8 ⊢ ((⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
6	5	ex 113	. . . . . . 7 ⊢ (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ → (𝜑 → ∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
7		vex 2622	. . . . . . . . 9 ⊢ 𝑧 ∈ V
8		vex 2622	. . . . . . . . 9 ⊢ 𝑤 ∈ V
9	7, 8	opth 4055	. . . . . . . 8 ⊢ (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑧 = 𝑥 ∧ 𝑤 = 𝑦))
10	9	anbi1i 446	. . . . . . . . . 10 ⊢ ((⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑))
11	10	2exbii 1542	. . . . . . . . 9 ⊢ (∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑))
12		nfe1 1430	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∃𝑥(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑))
13		dveeq2or 1744	. . . . . . . . . . . 12 ⊢ (∀𝑦 𝑦 = 𝑥 ∨ Ⅎ𝑦 𝑧 = 𝑥)
14		nfae 1654	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑦∀𝑦 𝑦 = 𝑥
15		anass 393	. . . . . . . . . . . . . . . 16 ⊢ (((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) ↔ (𝑧 = 𝑥 ∧ (𝑤 = 𝑦 ∧ 𝜑)))
16		19.8a 1527	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 = 𝑦 ∧ 𝜑) → ∃𝑦(𝑤 = 𝑦 ∧ 𝜑))
17	16	a1i 9	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑦 𝑦 = 𝑥 → ((𝑤 = 𝑦 ∧ 𝜑) → ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)))
18	17	anim2d 330	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑦 𝑦 = 𝑥 → ((𝑧 = 𝑥 ∧ (𝑤 = 𝑦 ∧ 𝜑)) → (𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑))))
19	15, 18	syl5bi 150	. . . . . . . . . . . . . . 15 ⊢ (∀𝑦 𝑦 = 𝑥 → (((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) → (𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑))))
20	14, 19	eximd 1548	. . . . . . . . . . . . . 14 ⊢ (∀𝑦 𝑦 = 𝑥 → (∃𝑦((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) → ∃𝑦(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑))))
21		biidd 170	. . . . . . . . . . . . . . 15 ⊢ (∀𝑦 𝑦 = 𝑥 → ((𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)) ↔ (𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑))))
22	21	drex1 1726	. . . . . . . . . . . . . 14 ⊢ (∀𝑦 𝑦 = 𝑥 → (∃𝑦(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)) ↔ ∃𝑥(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑))))
23	20, 22	sylibd 147	. . . . . . . . . . . . 13 ⊢ (∀𝑦 𝑦 = 𝑥 → (∃𝑦((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) → ∃𝑥(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑))))
24	15	exbii 1541	. . . . . . . . . . . . . . 15 ⊢ (∃𝑦((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) ↔ ∃𝑦(𝑧 = 𝑥 ∧ (𝑤 = 𝑦 ∧ 𝜑)))
25		19.40 1567	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑦(𝑧 = 𝑥 ∧ (𝑤 = 𝑦 ∧ 𝜑)) → (∃𝑦 𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)))
26		19.9t 1578	. . . . . . . . . . . . . . . . . 18 ⊢ (Ⅎ𝑦 𝑧 = 𝑥 → (∃𝑦 𝑧 = 𝑥 ↔ 𝑧 = 𝑥))
27	26	biimpd 142	. . . . . . . . . . . . . . . . 17 ⊢ (Ⅎ𝑦 𝑧 = 𝑥 → (∃𝑦 𝑧 = 𝑥 → 𝑧 = 𝑥))
28	27	anim1d 329	. . . . . . . . . . . . . . . 16 ⊢ (Ⅎ𝑦 𝑧 = 𝑥 → ((∃𝑦 𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)) → (𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑))))
29	25, 28	syl5 32	. . . . . . . . . . . . . . 15 ⊢ (Ⅎ𝑦 𝑧 = 𝑥 → (∃𝑦(𝑧 = 𝑥 ∧ (𝑤 = 𝑦 ∧ 𝜑)) → (𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑))))
30	24, 29	syl5bi 150	. . . . . . . . . . . . . 14 ⊢ (Ⅎ𝑦 𝑧 = 𝑥 → (∃𝑦((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) → (𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑))))
31		19.8a 1527	. . . . . . . . . . . . . 14 ⊢ ((𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)) → ∃𝑥(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)))
32	30, 31	syl6 33	. . . . . . . . . . . . 13 ⊢ (Ⅎ𝑦 𝑧 = 𝑥 → (∃𝑦((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) → ∃𝑥(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑))))
33	23, 32	jaoi 671	. . . . . . . . . . . 12 ⊢ ((∀𝑦 𝑦 = 𝑥 ∨ Ⅎ𝑦 𝑧 = 𝑥) → (∃𝑦((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) → ∃𝑥(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑))))
34	13, 33	ax-mp 7	. . . . . . . . . . 11 ⊢ (∃𝑦((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) → ∃𝑥(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)))
35	12, 34	exlimi 1530	. . . . . . . . . 10 ⊢ (∃𝑥∃𝑦((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) → ∃𝑥(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)))
36		euequ1 2043	. . . . . . . . . . . . . 14 ⊢ ∃!𝑥 𝑥 = 𝑧
37		equcom 1639	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 ↔ 𝑧 = 𝑥)
38	37	eubii 1957	. . . . . . . . . . . . . 14 ⊢ (∃!𝑥 𝑥 = 𝑧 ↔ ∃!𝑥 𝑧 = 𝑥)
39	36, 38	mpbi 143	. . . . . . . . . . . . 13 ⊢ ∃!𝑥 𝑧 = 𝑥
40		eupick 2027	. . . . . . . . . . . . 13 ⊢ ((∃!𝑥 𝑧 = 𝑥 ∧ ∃𝑥(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑))) → (𝑧 = 𝑥 → ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)))
41	39, 40	mpan 415	. . . . . . . . . . . 12 ⊢ (∃𝑥(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)) → (𝑧 = 𝑥 → ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)))
42	41	com12 30	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (∃𝑥(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)) → ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)))
43		euequ1 2043	. . . . . . . . . . . . . 14 ⊢ ∃!𝑦 𝑦 = 𝑤
44		equcom 1639	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑤 ↔ 𝑤 = 𝑦)
45	44	eubii 1957	. . . . . . . . . . . . . 14 ⊢ (∃!𝑦 𝑦 = 𝑤 ↔ ∃!𝑦 𝑤 = 𝑦)
46	43, 45	mpbi 143	. . . . . . . . . . . . 13 ⊢ ∃!𝑦 𝑤 = 𝑦
47		eupick 2027	. . . . . . . . . . . . 13 ⊢ ((∃!𝑦 𝑤 = 𝑦 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)) → (𝑤 = 𝑦 → 𝜑))
48	46, 47	mpan 415	. . . . . . . . . . . 12 ⊢ (∃𝑦(𝑤 = 𝑦 ∧ 𝜑) → (𝑤 = 𝑦 → 𝜑))
49	48	com12 30	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑦 → (∃𝑦(𝑤 = 𝑦 ∧ 𝜑) → 𝜑))
50	42, 49	sylan9 401	. . . . . . . . . 10 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → (∃𝑥(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)) → 𝜑))
51	35, 50	syl5 32	. . . . . . . . 9 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → (∃𝑥∃𝑦((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) → 𝜑))
52	11, 51	syl5bi 150	. . . . . . . 8 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → (∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝜑))
53	9, 52	sylbi 119	. . . . . . 7 ⊢ (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ → (∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝜑))
54	6, 53	impbid 127	. . . . . 6 ⊢ (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
55		eqeq1 2094	. . . . . . 7 ⊢ (𝐴 = ⟨𝑧, 𝑤⟩ → (𝐴 = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩))
56	55	anbi1d 453	. . . . . . . . 9 ⊢ (𝐴 = ⟨𝑧, 𝑤⟩ → ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
57	56	2exbidv 1796	. . . . . . . 8 ⊢ (𝐴 = ⟨𝑧, 𝑤⟩ → (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
58	57	bibi2d 230	. . . . . . 7 ⊢ (𝐴 = ⟨𝑧, 𝑤⟩ → ((𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ (𝜑 ↔ ∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))))
59	55, 58	imbi12d 232	. . . . . 6 ⊢ (𝐴 = ⟨𝑧, 𝑤⟩ → ((𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))) ↔ (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))))
60	54, 59	mpbiri 166	. . . . 5 ⊢ (𝐴 = ⟨𝑧, 𝑤⟩ → (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))))
61	60	adantr 270	. . . 4 ⊢ ((𝐴 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩) → (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))))
62	61	exlimivv 1824	. . 3 ⊢ (∃𝑧∃𝑤(𝐴 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩) → (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))))
63	3, 62	sylbi 119	. 2 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))))
64	63	pm2.43i 48	1 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   ∨ wo 664  ∀wal 1287   = wceq 1289  Ⅎwnf 1394  ∃wex 1426  ∃!weu 1948  ⟨cop 3444

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450

This theorem is referenced by:  copsex2t  4063  copsex2g  4064  opabid  4075  mosubopt  4491