Theorem ich2exprop 48075

Description: If the setvar variables are interchangeable in a wff, there is an ordered pair fulfilling the wff iff there is an unordered pair fulfilling the wff. (Contributed by AV, 16-Jul-2023.)

Assertion

Ref	Expression
ich2exprop	⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ [𝑎⇄𝑏]𝜑) → (∃𝑎∃𝑏({𝐴, 𝐵} = {𝑎, 𝑏} ∧ 𝜑) ↔ ∃𝑎∃𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)))

Distinct variable groups:   𝐴,𝑎,𝑏   𝐵,𝑎,𝑏   𝑋,𝑎,𝑏

Allowed substitution hints:   𝜑(𝑎,𝑏)

Proof of Theorem ich2exprop

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1937	. . . . 5 ⊢ Ⅎ𝑎 𝐴 ∈ 𝑋
2		nfv 1937	. . . . 5 ⊢ Ⅎ𝑎 𝐵 ∈ 𝑋
3		nfich1 48051	. . . . 5 ⊢ Ⅎ𝑎[𝑎⇄𝑏]𝜑
4	1, 2, 3	nf3an 1924	. . . 4 ⊢ Ⅎ𝑎(𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ [𝑎⇄𝑏]𝜑)
5		nfv 1937	. . . . . . 7 ⊢ Ⅎ𝑎⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩
6		nfcv 2927	. . . . . . . 8 ⊢ Ⅎ𝑎𝑦
7		nfsbc1v 3767	. . . . . . . 8 ⊢ Ⅎ𝑎[𝑥 / 𝑎]𝜑
8	6, 7	nfsbcw 3769	. . . . . . 7 ⊢ Ⅎ𝑎[𝑦 / 𝑏][𝑥 / 𝑎]𝜑
9	5, 8	nfan 1922	. . . . . 6 ⊢ Ⅎ𝑎(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑)
10	9	nfex 2359	. . . . 5 ⊢ Ⅎ𝑎∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑)
11	10	nfex 2359	. . . 4 ⊢ Ⅎ𝑎∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑)
12		nfv 1937	. . . . . 6 ⊢ Ⅎ𝑏 𝐴 ∈ 𝑋
13		nfv 1937	. . . . . 6 ⊢ Ⅎ𝑏 𝐵 ∈ 𝑋
14		nfich2 48052	. . . . . 6 ⊢ Ⅎ𝑏[𝑎⇄𝑏]𝜑
15	12, 13, 14	nf3an 1924	. . . . 5 ⊢ Ⅎ𝑏(𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ [𝑎⇄𝑏]𝜑)
16		nfv 1937	. . . . . . . 8 ⊢ Ⅎ𝑏⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩
17		nfsbc1v 3767	. . . . . . . 8 ⊢ Ⅎ𝑏[𝑦 / 𝑏][𝑥 / 𝑎]𝜑
18	16, 17	nfan 1922	. . . . . . 7 ⊢ Ⅎ𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑)
19	18	nfex 2359	. . . . . 6 ⊢ Ⅎ𝑏∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑)
20	19	nfex 2359	. . . . 5 ⊢ Ⅎ𝑏∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑)
21		vex 3461	. . . . . . . . 9 ⊢ 𝑎 ∈ V
22		vex 3461	. . . . . . . . 9 ⊢ 𝑏 ∈ V
23		preq12bg 4814	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑎 ∈ V ∧ 𝑏 ∈ V)) → ({𝐴, 𝐵} = {𝑎, 𝑏} ↔ ((𝐴 = 𝑎 ∧ 𝐵 = 𝑏) ∨ (𝐴 = 𝑏 ∧ 𝐵 = 𝑎))))
24	21, 22, 23	mpanr12 717	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ({𝐴, 𝐵} = {𝑎, 𝑏} ↔ ((𝐴 = 𝑎 ∧ 𝐵 = 𝑏) ∨ (𝐴 = 𝑏 ∧ 𝐵 = 𝑎))))
25	24	3adant3 1148	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ [𝑎⇄𝑏]𝜑) → ({𝐴, 𝐵} = {𝑎, 𝑏} ↔ ((𝐴 = 𝑎 ∧ 𝐵 = 𝑏) ∨ (𝐴 = 𝑏 ∧ 𝐵 = 𝑎))))
26		or2expropbilem1 47624	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 = 𝑎 ∧ 𝐵 = 𝑏) → (𝜑 → ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑))))
27	26	3adant3 1148	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ [𝑎⇄𝑏]𝜑) → ((𝐴 = 𝑎 ∧ 𝐵 = 𝑏) → (𝜑 → ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑))))
28		ichcom 48063	. . . . . . . . . . . . . . 15 ⊢ ([𝑎⇄𝑏]𝜑 ↔ [𝑏⇄𝑎]𝜑)
29	28	biimpi 219	. . . . . . . . . . . . . 14 ⊢ ([𝑎⇄𝑏]𝜑 → [𝑏⇄𝑎]𝜑)
30	29	3ad2ant3 1151	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ [𝑎⇄𝑏]𝜑) → [𝑏⇄𝑎]𝜑)
31	30	adantr 485	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ [𝑎⇄𝑏]𝜑) ∧ 𝜑) → [𝑏⇄𝑎]𝜑)
32	22, 21	pm3.2i 475	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ V ∧ 𝑎 ∈ V)
33	32	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐴 = 𝑏 ∧ 𝐵 = 𝑎) → (𝑏 ∈ V ∧ 𝑎 ∈ V))
34	31, 33	anim12i 624	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ [𝑎⇄𝑏]𝜑) ∧ 𝜑) ∧ (𝐴 = 𝑏 ∧ 𝐵 = 𝑎)) → ([𝑏⇄𝑎]𝜑 ∧ (𝑏 ∈ V ∧ 𝑎 ∈ V)))
35		simpr 489	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ [𝑎⇄𝑏]𝜑) ∧ 𝜑) → 𝜑)
36		opeq12 4836	. . . . . . . . . . . 12 ⊢ ((𝐴 = 𝑏 ∧ 𝐵 = 𝑎) → ⟨𝐴, 𝐵⟩ = ⟨𝑏, 𝑎⟩)
37	35, 36	anim12ci 625	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ [𝑎⇄𝑏]𝜑) ∧ 𝜑) ∧ (𝐴 = 𝑏 ∧ 𝐵 = 𝑎)) → (⟨𝐴, 𝐵⟩ = ⟨𝑏, 𝑎⟩ ∧ 𝜑))
38		nfv 1937	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(⟨𝐴, 𝐵⟩ = ⟨𝑏, 𝑎⟩ ∧ 𝜑)
39		nfv 1937	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑏, 𝑎⟩ ∧ 𝜑)
40		opeq12 4836	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = 𝑏 ∧ 𝑦 = 𝑎) → ⟨𝑥, 𝑦⟩ = ⟨𝑏, 𝑎⟩)
41	40	eqeq2d 2776	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝑏 ∧ 𝑦 = 𝑎) → (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑏, 𝑎⟩))
42	41	adantl 486	. . . . . . . . . . . . 13 ⊢ (([𝑏⇄𝑎]𝜑 ∧ (𝑥 = 𝑏 ∧ 𝑦 = 𝑎)) → (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑏, 𝑎⟩))
43		dfsbcq 3749	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑎 → ([𝑦 / 𝑏][𝑥 / 𝑎]𝜑 ↔ [𝑎 / 𝑏][𝑥 / 𝑎]𝜑))
44	43	adantl 486	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = 𝑏 ∧ 𝑦 = 𝑎) → ([𝑦 / 𝑏][𝑥 / 𝑎]𝜑 ↔ [𝑎 / 𝑏][𝑥 / 𝑎]𝜑))
45	44	adantl 486	. . . . . . . . . . . . . 14 ⊢ (([𝑏⇄𝑎]𝜑 ∧ (𝑥 = 𝑏 ∧ 𝑦 = 𝑎)) → ([𝑦 / 𝑏][𝑥 / 𝑎]𝜑 ↔ [𝑎 / 𝑏][𝑥 / 𝑎]𝜑))
46		sbceq1a 3758	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑏 → ([𝑎 / 𝑏][𝑥 / 𝑎]𝜑 ↔ [𝑏 / 𝑥][𝑎 / 𝑏][𝑥 / 𝑎]𝜑))
47	46	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = 𝑏 ∧ 𝑦 = 𝑎) → ([𝑎 / 𝑏][𝑥 / 𝑎]𝜑 ↔ [𝑏 / 𝑥][𝑎 / 𝑏][𝑥 / 𝑎]𝜑))
48		df-ich 48050	. . . . . . . . . . . . . . . 16 ⊢ ([𝑏⇄𝑎]𝜑 ↔ ∀𝑏∀𝑎([𝑏 / 𝑥][𝑎 / 𝑏][𝑥 / 𝑎]𝜑 ↔ 𝜑))
49		sbsbc 3751	. . . . . . . . . . . . . . . . . 18 ⊢ ([𝑏 / 𝑥][𝑎 / 𝑏][𝑥 / 𝑎]𝜑 ↔ [𝑏 / 𝑥][𝑎 / 𝑏][𝑥 / 𝑎]𝜑)
50		sbsbc 3751	. . . . . . . . . . . . . . . . . . . 20 ⊢ ([𝑎 / 𝑏][𝑥 / 𝑎]𝜑 ↔ [𝑎 / 𝑏][𝑥 / 𝑎]𝜑)
51		sbsbc 3751	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ([𝑥 / 𝑎]𝜑 ↔ [𝑥 / 𝑎]𝜑)
52	51	sbcbii 3803	. . . . . . . . . . . . . . . . . . . 20 ⊢ ([𝑎 / 𝑏][𝑥 / 𝑎]𝜑 ↔ [𝑎 / 𝑏][𝑥 / 𝑎]𝜑)
53	50, 52	bitri 278	. . . . . . . . . . . . . . . . . . 19 ⊢ ([𝑎 / 𝑏][𝑥 / 𝑎]𝜑 ↔ [𝑎 / 𝑏][𝑥 / 𝑎]𝜑)
54	53	sbcbii 3803	. . . . . . . . . . . . . . . . . 18 ⊢ ([𝑏 / 𝑥][𝑎 / 𝑏][𝑥 / 𝑎]𝜑 ↔ [𝑏 / 𝑥][𝑎 / 𝑏][𝑥 / 𝑎]𝜑)
55	49, 54	bitri 278	. . . . . . . . . . . . . . . . 17 ⊢ ([𝑏 / 𝑥][𝑎 / 𝑏][𝑥 / 𝑎]𝜑 ↔ [𝑏 / 𝑥][𝑎 / 𝑏][𝑥 / 𝑎]𝜑)
56		2sp 2224	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑏∀𝑎([𝑏 / 𝑥][𝑎 / 𝑏][𝑥 / 𝑎]𝜑 ↔ 𝜑) → ([𝑏 / 𝑥][𝑎 / 𝑏][𝑥 / 𝑎]𝜑 ↔ 𝜑))
57	55, 56	bitr3id 288	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑏∀𝑎([𝑏 / 𝑥][𝑎 / 𝑏][𝑥 / 𝑎]𝜑 ↔ 𝜑) → ([𝑏 / 𝑥][𝑎 / 𝑏][𝑥 / 𝑎]𝜑 ↔ 𝜑))
58	48, 57	sylbi 220	. . . . . . . . . . . . . . 15 ⊢ ([𝑏⇄𝑎]𝜑 → ([𝑏 / 𝑥][𝑎 / 𝑏][𝑥 / 𝑎]𝜑 ↔ 𝜑))
59	47, 58	sylan9bbr 519	. . . . . . . . . . . . . 14 ⊢ (([𝑏⇄𝑎]𝜑 ∧ (𝑥 = 𝑏 ∧ 𝑦 = 𝑎)) → ([𝑎 / 𝑏][𝑥 / 𝑎]𝜑 ↔ 𝜑))
60	45, 59	bitrd 282	. . . . . . . . . . . . 13 ⊢ (([𝑏⇄𝑎]𝜑 ∧ (𝑥 = 𝑏 ∧ 𝑦 = 𝑎)) → ([𝑦 / 𝑏][𝑥 / 𝑎]𝜑 ↔ 𝜑))
61	42, 60	anbi12d 643	. . . . . . . . . . . 12 ⊢ (([𝑏⇄𝑎]𝜑 ∧ (𝑥 = 𝑏 ∧ 𝑦 = 𝑎)) → ((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑏, 𝑎⟩ ∧ 𝜑)))
62	38, 39, 61	spc2ed 3563	. . . . . . . . . . 11 ⊢ (([𝑏⇄𝑎]𝜑 ∧ (𝑏 ∈ V ∧ 𝑎 ∈ V)) → ((⟨𝐴, 𝐵⟩ = ⟨𝑏, 𝑎⟩ ∧ 𝜑) → ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑)))
63	34, 37, 62	sylc 66	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ [𝑎⇄𝑏]𝜑) ∧ 𝜑) ∧ (𝐴 = 𝑏 ∧ 𝐵 = 𝑎)) → ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑))
64	63	exp31 424	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ [𝑎⇄𝑏]𝜑) → (𝜑 → ((𝐴 = 𝑏 ∧ 𝐵 = 𝑎) → ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑))))
65	64	com23 87	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ [𝑎⇄𝑏]𝜑) → ((𝐴 = 𝑏 ∧ 𝐵 = 𝑎) → (𝜑 → ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑))))
66	27, 65	jaod 872	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ [𝑎⇄𝑏]𝜑) → (((𝐴 = 𝑎 ∧ 𝐵 = 𝑏) ∨ (𝐴 = 𝑏 ∧ 𝐵 = 𝑎)) → (𝜑 → ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑))))
67	25, 66	sylbid 243	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ [𝑎⇄𝑏]𝜑) → ({𝐴, 𝐵} = {𝑎, 𝑏} → (𝜑 → ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑))))
68	67	impd 415	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ [𝑎⇄𝑏]𝜑) → (({𝐴, 𝐵} = {𝑎, 𝑏} ∧ 𝜑) → ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑)))
69	15, 20, 68	exlimd 2256	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ [𝑎⇄𝑏]𝜑) → (∃𝑏({𝐴, 𝐵} = {𝑎, 𝑏} ∧ 𝜑) → ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑)))
70	4, 11, 69	exlimd 2256	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ [𝑎⇄𝑏]𝜑) → (∃𝑎∃𝑏({𝐴, 𝐵} = {𝑎, 𝑏} ∧ 𝜑) → ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑)))
71		or2expropbilem2 47625	. . 3 ⊢ (∃𝑎∃𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑))
72	70, 71	imbitrrdi 255	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ [𝑎⇄𝑏]𝜑) → (∃𝑎∃𝑏({𝐴, 𝐵} = {𝑎, 𝑏} ∧ 𝜑) → ∃𝑎∃𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)))
73		oppr 47622	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ → {𝐴, 𝐵} = {𝑎, 𝑏}))
74	73	anim1d 622	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ({𝐴, 𝐵} = {𝑎, 𝑏} ∧ 𝜑)))
75	74	2eximdv 1942	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∃𝑎∃𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ∃𝑎∃𝑏({𝐴, 𝐵} = {𝑎, 𝑏} ∧ 𝜑)))
76	75	3adant3 1148	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ [𝑎⇄𝑏]𝜑) → (∃𝑎∃𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ∃𝑎∃𝑏({𝐴, 𝐵} = {𝑎, 𝑏} ∧ 𝜑)))
77	72, 76	impbid 215	1 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ [𝑎⇄𝑏]𝜑) → (∃𝑎∃𝑏({𝐴, 𝐵} = {𝑎, 𝑏} ∧ 𝜑) ↔ ∃𝑎∃𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1101  ∀wal 1561   = wceq 1563  ∃wex 1802  [wsb 2093   ∈ wcel 2145  Vcvv 3457  [wsbc 3747  {cpr 4587  ⟨cop 4591  [wich 48049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-rab 3418  df-v 3459  df-sbc 3748  df-dif 3910  df-un 3912  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-ich 48050

This theorem is referenced by: (None)