Theorem ichreuopeq 44936

Description: If the setvar variables are interchangeable in a wff, and there is a unique ordered pair fulfilling the wff, then both setvar variables must be equal. (Contributed by AV, 28-Aug-2023.)

Assertion

Ref	Expression
ichreuopeq	⊢ ([𝑎⇄𝑏]𝜑 → (∃!𝑝 ∈ (𝑋 × 𝑋)∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ∃𝑎∃𝑏(𝑎 = 𝑏 ∧ 𝜑)))

Distinct variable groups:   𝑋,𝑎,𝑏,𝑝   𝜑,𝑝

Allowed substitution hints:   𝜑(𝑎,𝑏)

Proof of Theorem ichreuopeq

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

Step	Hyp	Ref	Expression
1		eqeq1 2743	. . . . 5 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝑝 = ⟨𝑎, 𝑏⟩ ↔ ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩))
2	1	anbi1d 630	. . . 4 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ↔ (⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)))
3	2	2exbidv 1928	. . 3 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ↔ ∃𝑎∃𝑏(⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)))
4		eqeq1 2743	. . . . 5 ⊢ (𝑝 = ⟨𝑣, 𝑤⟩ → (𝑝 = ⟨𝑎, 𝑏⟩ ↔ ⟨𝑣, 𝑤⟩ = ⟨𝑎, 𝑏⟩))
5	4	anbi1d 630	. . . 4 ⊢ (𝑝 = ⟨𝑣, 𝑤⟩ → ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ↔ (⟨𝑣, 𝑤⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)))
6	5	2exbidv 1928	. . 3 ⊢ (𝑝 = ⟨𝑣, 𝑤⟩ → (∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ↔ ∃𝑎∃𝑏(⟨𝑣, 𝑤⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)))
7	3, 6	reuop 6200	. 2 ⊢ (∃!𝑝 ∈ (𝑋 × 𝑋)∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ↔ ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ∧ ∀𝑣 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑣, 𝑤⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑣, 𝑤⟩ = ⟨𝑥, 𝑦⟩)))
8		nfich1 44910	. . . . . 6 ⊢ Ⅎ𝑎[𝑎⇄𝑏]𝜑
9		nfv 1918	. . . . . 6 ⊢ Ⅎ𝑎(𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)
10	8, 9	nfan 1903	. . . . 5 ⊢ Ⅎ𝑎([𝑎⇄𝑏]𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋))
11		nfcv 2908	. . . . . . 7 ⊢ Ⅎ𝑎𝑋
12		nfe1 2148	. . . . . . . . 9 ⊢ Ⅎ𝑎∃𝑎∃𝑏(⟨𝑣, 𝑤⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)
13		nfv 1918	. . . . . . . . 9 ⊢ Ⅎ𝑎⟨𝑣, 𝑤⟩ = ⟨𝑥, 𝑦⟩
14	12, 13	nfim 1900	. . . . . . . 8 ⊢ Ⅎ𝑎(∃𝑎∃𝑏(⟨𝑣, 𝑤⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑣, 𝑤⟩ = ⟨𝑥, 𝑦⟩)
15	11, 14	nfralw 3152	. . . . . . 7 ⊢ Ⅎ𝑎∀𝑤 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑣, 𝑤⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑣, 𝑤⟩ = ⟨𝑥, 𝑦⟩)
16	11, 15	nfralw 3152	. . . . . 6 ⊢ Ⅎ𝑎∀𝑣 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑣, 𝑤⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑣, 𝑤⟩ = ⟨𝑥, 𝑦⟩)
17		nfe1 2148	. . . . . 6 ⊢ Ⅎ𝑎∃𝑎∃𝑏(𝑎 = 𝑏 ∧ 𝜑)
18	16, 17	nfim 1900	. . . . 5 ⊢ Ⅎ𝑎(∀𝑣 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑣, 𝑤⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑣, 𝑤⟩ = ⟨𝑥, 𝑦⟩) → ∃𝑎∃𝑏(𝑎 = 𝑏 ∧ 𝜑))
19		nfich2 44911	. . . . . . 7 ⊢ Ⅎ𝑏[𝑎⇄𝑏]𝜑
20		nfv 1918	. . . . . . 7 ⊢ Ⅎ𝑏(𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)
21	19, 20	nfan 1903	. . . . . 6 ⊢ Ⅎ𝑏([𝑎⇄𝑏]𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋))
22		nfcv 2908	. . . . . . . 8 ⊢ Ⅎ𝑏𝑋
23		nfe1 2148	. . . . . . . . . . 11 ⊢ Ⅎ𝑏∃𝑏(⟨𝑣, 𝑤⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)
24	23	nfex 2319	. . . . . . . . . 10 ⊢ Ⅎ𝑏∃𝑎∃𝑏(⟨𝑣, 𝑤⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)
25		nfv 1918	. . . . . . . . . 10 ⊢ Ⅎ𝑏⟨𝑣, 𝑤⟩ = ⟨𝑥, 𝑦⟩
26	24, 25	nfim 1900	. . . . . . . . 9 ⊢ Ⅎ𝑏(∃𝑎∃𝑏(⟨𝑣, 𝑤⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑣, 𝑤⟩ = ⟨𝑥, 𝑦⟩)
27	22, 26	nfralw 3152	. . . . . . . 8 ⊢ Ⅎ𝑏∀𝑤 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑣, 𝑤⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑣, 𝑤⟩ = ⟨𝑥, 𝑦⟩)
28	22, 27	nfralw 3152	. . . . . . 7 ⊢ Ⅎ𝑏∀𝑣 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑣, 𝑤⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑣, 𝑤⟩ = ⟨𝑥, 𝑦⟩)
29		nfe1 2148	. . . . . . . 8 ⊢ Ⅎ𝑏∃𝑏(𝑎 = 𝑏 ∧ 𝜑)
30	29	nfex 2319	. . . . . . 7 ⊢ Ⅎ𝑏∃𝑎∃𝑏(𝑎 = 𝑏 ∧ 𝜑)
31	28, 30	nfim 1900	. . . . . 6 ⊢ Ⅎ𝑏(∀𝑣 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑣, 𝑤⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑣, 𝑤⟩ = ⟨𝑥, 𝑦⟩) → ∃𝑎∃𝑏(𝑎 = 𝑏 ∧ 𝜑))
32		opeq12 4807	. . . . . . . . . . . . . 14 ⊢ ((𝑣 = 𝑦 ∧ 𝑤 = 𝑥) → ⟨𝑣, 𝑤⟩ = ⟨𝑦, 𝑥⟩)
33	32	eqeq1d 2741	. . . . . . . . . . . . 13 ⊢ ((𝑣 = 𝑦 ∧ 𝑤 = 𝑥) → (⟨𝑣, 𝑤⟩ = ⟨𝑎, 𝑏⟩ ↔ ⟨𝑦, 𝑥⟩ = ⟨𝑎, 𝑏⟩))
34	33	anbi1d 630	. . . . . . . . . . . 12 ⊢ ((𝑣 = 𝑦 ∧ 𝑤 = 𝑥) → ((⟨𝑣, 𝑤⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ↔ (⟨𝑦, 𝑥⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)))
35	34	2exbidv 1928	. . . . . . . . . . 11 ⊢ ((𝑣 = 𝑦 ∧ 𝑤 = 𝑥) → (∃𝑎∃𝑏(⟨𝑣, 𝑤⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ↔ ∃𝑎∃𝑏(⟨𝑦, 𝑥⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)))
36	32	eqeq1d 2741	. . . . . . . . . . 11 ⊢ ((𝑣 = 𝑦 ∧ 𝑤 = 𝑥) → (⟨𝑣, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑦, 𝑥⟩ = ⟨𝑥, 𝑦⟩))
37	35, 36	imbi12d 345	. . . . . . . . . 10 ⊢ ((𝑣 = 𝑦 ∧ 𝑤 = 𝑥) → ((∃𝑎∃𝑏(⟨𝑣, 𝑤⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑣, 𝑤⟩ = ⟨𝑥, 𝑦⟩) ↔ (∃𝑎∃𝑏(⟨𝑦, 𝑥⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑦, 𝑥⟩ = ⟨𝑥, 𝑦⟩)))
38	37	rspc2gv 3570	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (∀𝑣 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑣, 𝑤⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑣, 𝑤⟩ = ⟨𝑥, 𝑦⟩) → (∃𝑎∃𝑏(⟨𝑦, 𝑥⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑦, 𝑥⟩ = ⟨𝑥, 𝑦⟩)))
39	38	ancoms 459	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (∀𝑣 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑣, 𝑤⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑣, 𝑤⟩ = ⟨𝑥, 𝑦⟩) → (∃𝑎∃𝑏(⟨𝑦, 𝑥⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑦, 𝑥⟩ = ⟨𝑥, 𝑦⟩)))
40	39	adantl 482	. . . . . . 7 ⊢ (([𝑎⇄𝑏]𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (∀𝑣 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑣, 𝑤⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑣, 𝑤⟩ = ⟨𝑥, 𝑦⟩) → (∃𝑎∃𝑏(⟨𝑦, 𝑥⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑦, 𝑥⟩ = ⟨𝑥, 𝑦⟩)))
41		simprr 770	. . . . . . . . . . 11 ⊢ (([𝑎⇄𝑏]𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
42	41	adantr 481	. . . . . . . . . 10 ⊢ ((([𝑎⇄𝑏]𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ (⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)) → 𝑦 ∈ 𝑋)
43		simpl 483	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 𝑥 ∈ 𝑋)
44	43	adantl 482	. . . . . . . . . . . 12 ⊢ (([𝑎⇄𝑏]𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
45	44	adantr 481	. . . . . . . . . . 11 ⊢ ((([𝑎⇄𝑏]𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ (⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)) → 𝑥 ∈ 𝑋)
46		eqidd 2740	. . . . . . . . . . . 12 ⊢ ((([𝑎⇄𝑏]𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ (⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)) → ⟨𝑦, 𝑥⟩ = ⟨𝑦, 𝑥⟩)
47		vex 3437	. . . . . . . . . . . . . . . . 17 ⊢ 𝑥 ∈ V
48		vex 3437	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
49	47, 48	opth 5392	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ↔ (𝑥 = 𝑎 ∧ 𝑦 = 𝑏))
50		sbceq1a 3728	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑏]𝜑))
51	50	equcoms 2024	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑏 → (𝜑 ↔ [𝑦 / 𝑏]𝜑))
52		sbceq1a 3728	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑥 → ([𝑦 / 𝑏]𝜑 ↔ [𝑥 / 𝑎][𝑦 / 𝑏]𝜑))
53	52	equcoms 2024	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → ([𝑦 / 𝑏]𝜑 ↔ [𝑥 / 𝑎][𝑦 / 𝑏]𝜑))
54	51, 53	sylan9bbr 511	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝜑 ↔ [𝑥 / 𝑎][𝑦 / 𝑏]𝜑))
55		dfich2 44921	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ([𝑎⇄𝑏]𝜑 ↔ ∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑏]𝜑 ↔ [𝑦 / 𝑎][𝑥 / 𝑏]𝜑))
56		2sp 2180	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑏]𝜑 ↔ [𝑦 / 𝑎][𝑥 / 𝑏]𝜑) → ([𝑥 / 𝑎][𝑦 / 𝑏]𝜑 ↔ [𝑦 / 𝑎][𝑥 / 𝑏]𝜑))
57		sbsbc 3721	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ([𝑦 / 𝑏]𝜑 ↔ [𝑦 / 𝑏]𝜑)
58	57	sbbii 2080	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ([𝑥 / 𝑎][𝑦 / 𝑏]𝜑 ↔ [𝑥 / 𝑎][𝑦 / 𝑏]𝜑)
59		sbsbc 3721	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ([𝑥 / 𝑎][𝑦 / 𝑏]𝜑 ↔ [𝑥 / 𝑎][𝑦 / 𝑏]𝜑)
60	58, 59	bitri 274	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ([𝑥 / 𝑎][𝑦 / 𝑏]𝜑 ↔ [𝑥 / 𝑎][𝑦 / 𝑏]𝜑)
61		sbsbc 3721	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ([𝑥 / 𝑏]𝜑 ↔ [𝑥 / 𝑏]𝜑)
62	61	sbbii 2080	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ([𝑦 / 𝑎][𝑥 / 𝑏]𝜑 ↔ [𝑦 / 𝑎][𝑥 / 𝑏]𝜑)
63		sbsbc 3721	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ([𝑦 / 𝑎][𝑥 / 𝑏]𝜑 ↔ [𝑦 / 𝑎][𝑥 / 𝑏]𝜑)
64	62, 63	bitri 274	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ([𝑦 / 𝑎][𝑥 / 𝑏]𝜑 ↔ [𝑦 / 𝑎][𝑥 / 𝑏]𝜑)
65	56, 60, 64	3bitr3g 313	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑏]𝜑 ↔ [𝑦 / 𝑎][𝑥 / 𝑏]𝜑) → ([𝑥 / 𝑎][𝑦 / 𝑏]𝜑 ↔ [𝑦 / 𝑎][𝑥 / 𝑏]𝜑))
66	55, 65	sylbi 216	. . . . . . . . . . . . . . . . . . . 20 ⊢ ([𝑎⇄𝑏]𝜑 → ([𝑥 / 𝑎][𝑦 / 𝑏]𝜑 ↔ [𝑦 / 𝑎][𝑥 / 𝑏]𝜑))
67	66	biimpd 228	. . . . . . . . . . . . . . . . . . 19 ⊢ ([𝑎⇄𝑏]𝜑 → ([𝑥 / 𝑎][𝑦 / 𝑏]𝜑 → [𝑦 / 𝑎][𝑥 / 𝑏]𝜑))
68	67	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (([𝑎⇄𝑏]𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ([𝑥 / 𝑎][𝑦 / 𝑏]𝜑 → [𝑦 / 𝑎][𝑥 / 𝑏]𝜑))
69	68	com12 32	. . . . . . . . . . . . . . . . 17 ⊢ ([𝑥 / 𝑎][𝑦 / 𝑏]𝜑 → (([𝑎⇄𝑏]𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → [𝑦 / 𝑎][𝑥 / 𝑏]𝜑))
70	54, 69	syl6bi 252	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝜑 → (([𝑎⇄𝑏]𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → [𝑦 / 𝑎][𝑥 / 𝑏]𝜑)))
71	49, 70	sylbi 216	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ → (𝜑 → (([𝑎⇄𝑏]𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → [𝑦 / 𝑎][𝑥 / 𝑏]𝜑)))
72	71	imp 407	. . . . . . . . . . . . . 14 ⊢ ((⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → (([𝑎⇄𝑏]𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → [𝑦 / 𝑎][𝑥 / 𝑏]𝜑))
73	72	impcom 408	. . . . . . . . . . . . 13 ⊢ ((([𝑎⇄𝑏]𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ (⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)) → [𝑦 / 𝑎][𝑥 / 𝑏]𝜑)
74		sbccom 3805	. . . . . . . . . . . . 13 ⊢ ([𝑥 / 𝑏][𝑦 / 𝑎]𝜑 ↔ [𝑦 / 𝑎][𝑥 / 𝑏]𝜑)
75	73, 74	sylibr 233	. . . . . . . . . . . 12 ⊢ ((([𝑎⇄𝑏]𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ (⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)) → [𝑥 / 𝑏][𝑦 / 𝑎]𝜑)
76	46, 75	jca 512	. . . . . . . . . . 11 ⊢ ((([𝑎⇄𝑏]𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ (⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)) → (⟨𝑦, 𝑥⟩ = ⟨𝑦, 𝑥⟩ ∧ [𝑥 / 𝑏][𝑦 / 𝑎]𝜑))
77		nfcv 2908	. . . . . . . . . . . 12 ⊢ Ⅎ𝑏𝑥
78		nfv 1918	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑏⟨𝑦, 𝑥⟩ = ⟨𝑦, 𝑥⟩
79		nfsbc1v 3737	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑏[𝑥 / 𝑏][𝑦 / 𝑎]𝜑
80	78, 79	nfan 1903	. . . . . . . . . . . 12 ⊢ Ⅎ𝑏(⟨𝑦, 𝑥⟩ = ⟨𝑦, 𝑥⟩ ∧ [𝑥 / 𝑏][𝑦 / 𝑎]𝜑)
81		opeq2 4806	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑥 → ⟨𝑦, 𝑏⟩ = ⟨𝑦, 𝑥⟩)
82	81	eqeq2d 2750	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑥 → (⟨𝑦, 𝑥⟩ = ⟨𝑦, 𝑏⟩ ↔ ⟨𝑦, 𝑥⟩ = ⟨𝑦, 𝑥⟩))
83		sbceq1a 3728	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑥 → ([𝑦 / 𝑎]𝜑 ↔ [𝑥 / 𝑏][𝑦 / 𝑎]𝜑))
84	82, 83	anbi12d 631	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑥 → ((⟨𝑦, 𝑥⟩ = ⟨𝑦, 𝑏⟩ ∧ [𝑦 / 𝑎]𝜑) ↔ (⟨𝑦, 𝑥⟩ = ⟨𝑦, 𝑥⟩ ∧ [𝑥 / 𝑏][𝑦 / 𝑎]𝜑)))
85	77, 80, 84	spcegf 3532	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑋 → ((⟨𝑦, 𝑥⟩ = ⟨𝑦, 𝑥⟩ ∧ [𝑥 / 𝑏][𝑦 / 𝑎]𝜑) → ∃𝑏(⟨𝑦, 𝑥⟩ = ⟨𝑦, 𝑏⟩ ∧ [𝑦 / 𝑎]𝜑)))
86	45, 76, 85	sylc 65	. . . . . . . . . 10 ⊢ ((([𝑎⇄𝑏]𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ (⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)) → ∃𝑏(⟨𝑦, 𝑥⟩ = ⟨𝑦, 𝑏⟩ ∧ [𝑦 / 𝑎]𝜑))
87		nfcv 2908	. . . . . . . . . . 11 ⊢ Ⅎ𝑎𝑦
88		nfv 1918	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑎⟨𝑦, 𝑥⟩ = ⟨𝑦, 𝑏⟩
89		nfsbc1v 3737	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑎[𝑦 / 𝑎]𝜑
90	88, 89	nfan 1903	. . . . . . . . . . . 12 ⊢ Ⅎ𝑎(⟨𝑦, 𝑥⟩ = ⟨𝑦, 𝑏⟩ ∧ [𝑦 / 𝑎]𝜑)
91	90	nfex 2319	. . . . . . . . . . 11 ⊢ Ⅎ𝑎∃𝑏(⟨𝑦, 𝑥⟩ = ⟨𝑦, 𝑏⟩ ∧ [𝑦 / 𝑎]𝜑)
92		opeq1 4805	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑦 → ⟨𝑎, 𝑏⟩ = ⟨𝑦, 𝑏⟩)
93	92	eqeq2d 2750	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑦 → (⟨𝑦, 𝑥⟩ = ⟨𝑎, 𝑏⟩ ↔ ⟨𝑦, 𝑥⟩ = ⟨𝑦, 𝑏⟩))
94		sbceq1a 3728	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑎]𝜑))
95	93, 94	anbi12d 631	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑦 → ((⟨𝑦, 𝑥⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ↔ (⟨𝑦, 𝑥⟩ = ⟨𝑦, 𝑏⟩ ∧ [𝑦 / 𝑎]𝜑)))
96	95	exbidv 1925	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑦 → (∃𝑏(⟨𝑦, 𝑥⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ↔ ∃𝑏(⟨𝑦, 𝑥⟩ = ⟨𝑦, 𝑏⟩ ∧ [𝑦 / 𝑎]𝜑)))
97	87, 91, 96	spcegf 3532	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑋 → (∃𝑏(⟨𝑦, 𝑥⟩ = ⟨𝑦, 𝑏⟩ ∧ [𝑦 / 𝑎]𝜑) → ∃𝑎∃𝑏(⟨𝑦, 𝑥⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)))
98	42, 86, 97	sylc 65	. . . . . . . . 9 ⊢ ((([𝑎⇄𝑏]𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ (⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)) → ∃𝑎∃𝑏(⟨𝑦, 𝑥⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑))
99		simpl 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 = 𝑥 ∧ (𝑥 = 𝑎 ∧ 𝑦 = 𝑏)) → 𝑦 = 𝑥)
100		simprr 770	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 = 𝑥 ∧ (𝑥 = 𝑎 ∧ 𝑦 = 𝑏)) → 𝑦 = 𝑏)
101		simpl 483	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → 𝑥 = 𝑎)
102	101	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 = 𝑥 ∧ (𝑥 = 𝑎 ∧ 𝑦 = 𝑏)) → 𝑥 = 𝑎)
103	99, 100, 102	3eqtr3rd 2788	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 = 𝑥 ∧ (𝑥 = 𝑎 ∧ 𝑦 = 𝑏)) → 𝑎 = 𝑏)
104	103	anim1i 615	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 = 𝑥 ∧ (𝑥 = 𝑎 ∧ 𝑦 = 𝑏)) ∧ 𝜑) → (𝑎 = 𝑏 ∧ 𝜑))
105	104	exp31 420	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑥 → ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝜑 → (𝑎 = 𝑏 ∧ 𝜑))))
106	49, 105	syl5bi 241	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑥 → (⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ → (𝜑 → (𝑎 = 𝑏 ∧ 𝜑))))
107	106	impd 411	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → ((⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → (𝑎 = 𝑏 ∧ 𝜑)))
108	48, 47	opth1 5391	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑦, 𝑥⟩ = ⟨𝑥, 𝑦⟩ → 𝑦 = 𝑥)
109	107, 108	syl11 33	. . . . . . . . . . . . . 14 ⊢ ((⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → (⟨𝑦, 𝑥⟩ = ⟨𝑥, 𝑦⟩ → (𝑎 = 𝑏 ∧ 𝜑)))
110	109	adantl 482	. . . . . . . . . . . . 13 ⊢ ((([𝑎⇄𝑏]𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ (⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)) → (⟨𝑦, 𝑥⟩ = ⟨𝑥, 𝑦⟩ → (𝑎 = 𝑏 ∧ 𝜑)))
111	110	imp 407	. . . . . . . . . . . 12 ⊢ (((([𝑎⇄𝑏]𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ (⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)) ∧ ⟨𝑦, 𝑥⟩ = ⟨𝑥, 𝑦⟩) → (𝑎 = 𝑏 ∧ 𝜑))
112	111	19.8ad 2176	. . . . . . . . . . 11 ⊢ (((([𝑎⇄𝑏]𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ (⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)) ∧ ⟨𝑦, 𝑥⟩ = ⟨𝑥, 𝑦⟩) → ∃𝑏(𝑎 = 𝑏 ∧ 𝜑))
113	112	19.8ad 2176	. . . . . . . . . 10 ⊢ (((([𝑎⇄𝑏]𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ (⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)) ∧ ⟨𝑦, 𝑥⟩ = ⟨𝑥, 𝑦⟩) → ∃𝑎∃𝑏(𝑎 = 𝑏 ∧ 𝜑))
114	113	ex 413	. . . . . . . . 9 ⊢ ((([𝑎⇄𝑏]𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ (⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)) → (⟨𝑦, 𝑥⟩ = ⟨𝑥, 𝑦⟩ → ∃𝑎∃𝑏(𝑎 = 𝑏 ∧ 𝜑)))
115	98, 114	embantd 59	. . . . . . . 8 ⊢ ((([𝑎⇄𝑏]𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ (⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)) → ((∃𝑎∃𝑏(⟨𝑦, 𝑥⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑦, 𝑥⟩ = ⟨𝑥, 𝑦⟩) → ∃𝑎∃𝑏(𝑎 = 𝑏 ∧ 𝜑)))
116	115	ex 413	. . . . . . 7 ⊢ (([𝑎⇄𝑏]𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ((∃𝑎∃𝑏(⟨𝑦, 𝑥⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑦, 𝑥⟩ = ⟨𝑥, 𝑦⟩) → ∃𝑎∃𝑏(𝑎 = 𝑏 ∧ 𝜑))))
117	40, 116	syl5d 73	. . . . . 6 ⊢ (([𝑎⇄𝑏]𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → (∀𝑣 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑣, 𝑤⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑣, 𝑤⟩ = ⟨𝑥, 𝑦⟩) → ∃𝑎∃𝑏(𝑎 = 𝑏 ∧ 𝜑))))
118	21, 31, 117	exlimd 2212	. . . . 5 ⊢ (([𝑎⇄𝑏]𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (∃𝑏(⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → (∀𝑣 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑣, 𝑤⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑣, 𝑤⟩ = ⟨𝑥, 𝑦⟩) → ∃𝑎∃𝑏(𝑎 = 𝑏 ∧ 𝜑))))
119	10, 18, 118	exlimd 2212	. . . 4 ⊢ (([𝑎⇄𝑏]𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (∃𝑎∃𝑏(⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → (∀𝑣 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑣, 𝑤⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑣, 𝑤⟩ = ⟨𝑥, 𝑦⟩) → ∃𝑎∃𝑏(𝑎 = 𝑏 ∧ 𝜑))))
120	119	impd 411	. . 3 ⊢ (([𝑎⇄𝑏]𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((∃𝑎∃𝑏(⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ∧ ∀𝑣 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑣, 𝑤⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑣, 𝑤⟩ = ⟨𝑥, 𝑦⟩)) → ∃𝑎∃𝑏(𝑎 = 𝑏 ∧ 𝜑)))
121	120	rexlimdvva 3224	. 2 ⊢ ([𝑎⇄𝑏]𝜑 → (∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ∧ ∀𝑣 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑣, 𝑤⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑣, 𝑤⟩ = ⟨𝑥, 𝑦⟩)) → ∃𝑎∃𝑏(𝑎 = 𝑏 ∧ 𝜑)))
122	7, 121	syl5bi 241	1 ⊢ ([𝑎⇄𝑏]𝜑 → (∃!𝑝 ∈ (𝑋 × 𝑋)∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ∃𝑎∃𝑏(𝑎 = 𝑏 ∧ 𝜑)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1537   = wceq 1539  ∃wex 1782  [wsb 2068   ∈ wcel 2107  ∀wral 3065  ∃wrex 3066  ∃!wreu 3067  [wsbc 3717  ⟨cop 4568   × cxp 5588  [wich 44908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2710  ax-sep 5224  ax-nul 5231  ax-pr 5353

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ral 3070  df-rex 3071  df-rmo 3072  df-reu 3073  df-rab 3074  df-v 3435  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-nul 4258  df-if 4461  df-sn 4563  df-pr 4565  df-op 4569  df-opab 5138  df-xp 5596  df-ich 44909

This theorem is referenced by: (None)