Theorem reupr 43733

Description: There is a unique unordered pair fulfilling a wff iff there are uniquely two sets fulfilling a corresponding wff. (Contributed by AV, 7-Apr-2023.)

Hypotheses

Ref	Expression
reupr.a	⊢ (𝑝 = {𝑎, 𝑏} → (𝜓 ↔ 𝜒))
reupr.x	⊢ (𝑝 = {𝑥, 𝑦} → (𝜓 ↔ 𝜃))

Assertion

Ref	Expression
reupr	⊢ (𝑋 ∈ 𝑉 → (∃!𝑝 ∈ (Pairs‘𝑋)𝜓 ↔ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝜒 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))))

Distinct variable groups:   𝑉,𝑎,𝑏,𝑝,𝑥,𝑦   𝑋,𝑎,𝑏,𝑝,𝑥,𝑦   𝜓,𝑎,𝑏,𝑥,𝑦   𝜃,𝑝   𝜒,𝑝

Allowed substitution hints:   𝜓(𝑝)   𝜒(𝑥,𝑦,𝑎,𝑏)   𝜃(𝑥,𝑦,𝑎,𝑏)

Proof of Theorem reupr

Dummy variables 𝑐 𝑑 𝑞 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfsbc1v 3792	. . 3 ⊢ Ⅎ𝑝[𝑞 / 𝑝]𝜓
2		nfsbc1v 3792	. . 3 ⊢ Ⅎ𝑝[𝑤 / 𝑝]𝜓
3		sbceq1a 3783	. . 3 ⊢ (𝑝 = 𝑤 → (𝜓 ↔ [𝑤 / 𝑝]𝜓))
4		dfsbcq 3774	. . 3 ⊢ (𝑤 = 𝑞 → ([𝑤 / 𝑝]𝜓 ↔ [𝑞 / 𝑝]𝜓))
5	1, 2, 3, 4	reu8nf 3860	. 2 ⊢ (∃!𝑝 ∈ (Pairs‘𝑋)𝜓 ↔ ∃𝑝 ∈ (Pairs‘𝑋)(𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → 𝑝 = 𝑞)))
6		sprel 43695	. . . . . 6 ⊢ (𝑝 ∈ (Pairs‘𝑋) → ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 𝑝 = {𝑎, 𝑏})
7		reupr.a	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = {𝑎, 𝑏} → (𝜓 ↔ 𝜒))
8	7	biimpcd 251	. . . . . . . . . . . . . 14 ⊢ (𝜓 → (𝑝 = {𝑎, 𝑏} → 𝜒))
9	8	adantr 483	. . . . . . . . . . . . 13 ⊢ ((𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → 𝑝 = 𝑞)) → (𝑝 = {𝑎, 𝑏} → 𝜒))
10	9	ad2antlr 725	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ 𝑉 ∧ (𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → 𝑝 = 𝑞))) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑝 = {𝑎, 𝑏} → 𝜒))
11	10	imp 409	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ 𝑉 ∧ (𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → 𝑝 = 𝑞))) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ 𝑝 = {𝑎, 𝑏}) → 𝜒)
12		pm3.22 462	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ 𝑋 ∈ 𝑉) → (𝑋 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)))
13	12	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ 𝑋 ∈ 𝑉) ∧ 𝜓) → (𝑋 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)))
14		prelspr 43697	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → {𝑥, 𝑦} ∈ (Pairs‘𝑋))
15		dfsbcq 3774	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑞 = {𝑥, 𝑦} → ([𝑞 / 𝑝]𝜓 ↔ [{𝑥, 𝑦} / 𝑝]𝜓))
16		eqeq2 2833	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑞 = {𝑥, 𝑦} → (𝑝 = 𝑞 ↔ 𝑝 = {𝑥, 𝑦}))
17	15, 16	imbi12d 347	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑞 = {𝑥, 𝑦} → (([𝑞 / 𝑝]𝜓 → 𝑝 = 𝑞) ↔ ([{𝑥, 𝑦} / 𝑝]𝜓 → 𝑝 = {𝑥, 𝑦})))
18	17	adantl 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑋 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝑞 = {𝑥, 𝑦}) → (([𝑞 / 𝑝]𝜓 → 𝑝 = 𝑞) ↔ ([{𝑥, 𝑦} / 𝑝]𝜓 → 𝑝 = {𝑥, 𝑦})))
19	14, 18	rspcdv 3615	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → 𝑝 = 𝑞) → ([{𝑥, 𝑦} / 𝑝]𝜓 → 𝑝 = {𝑥, 𝑦})))
20	13, 19	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ 𝑋 ∈ 𝑉) ∧ 𝜓) → (∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → 𝑝 = 𝑞) → ([{𝑥, 𝑦} / 𝑝]𝜓 → 𝑝 = {𝑥, 𝑦})))
21		zfpair2 5331	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ {𝑥, 𝑦} ∈ V
22		reupr.x	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑝 = {𝑥, 𝑦} → (𝜓 ↔ 𝜃))
23	21, 22	sbcie 3812	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ([{𝑥, 𝑦} / 𝑝]𝜓 ↔ 𝜃)
24		pm2.27 42	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ([{𝑥, 𝑦} / 𝑝]𝜓 → (([{𝑥, 𝑦} / 𝑝]𝜓 → 𝑝 = {𝑥, 𝑦}) → 𝑝 = {𝑥, 𝑦}))
25	23, 24	sylbir 237	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜃 → (([{𝑥, 𝑦} / 𝑝]𝜓 → 𝑝 = {𝑥, 𝑦}) → 𝑝 = {𝑥, 𝑦}))
26		eqcom 2828	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({𝑥, 𝑦} = 𝑝 ↔ 𝑝 = {𝑥, 𝑦})
27	25, 26	syl6ibr 254	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜃 → (([{𝑥, 𝑦} / 𝑝]𝜓 → 𝑝 = {𝑥, 𝑦}) → {𝑥, 𝑦} = 𝑝))
28	27	com12 32	. . . . . . . . . . . . . . . . . . 19 ⊢ (([{𝑥, 𝑦} / 𝑝]𝜓 → 𝑝 = {𝑥, 𝑦}) → (𝜃 → {𝑥, 𝑦} = 𝑝))
29		eqeq2 2833	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({𝑎, 𝑏} = 𝑝 → ({𝑥, 𝑦} = {𝑎, 𝑏} ↔ {𝑥, 𝑦} = 𝑝))
30	29	eqcoms 2829	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑝 = {𝑎, 𝑏} → ({𝑥, 𝑦} = {𝑎, 𝑏} ↔ {𝑥, 𝑦} = 𝑝))
31	30	imbi2d 343	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 = {𝑎, 𝑏} → ((𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}) ↔ (𝜃 → {𝑥, 𝑦} = 𝑝)))
32	28, 31	syl5ibrcom 249	. . . . . . . . . . . . . . . . . 18 ⊢ (([{𝑥, 𝑦} / 𝑝]𝜓 → 𝑝 = {𝑥, 𝑦}) → (𝑝 = {𝑎, 𝑏} → (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏})))
33	32	a1d 25	. . . . . . . . . . . . . . . . 17 ⊢ (([{𝑥, 𝑦} / 𝑝]𝜓 → 𝑝 = {𝑥, 𝑦}) → ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → (𝑝 = {𝑎, 𝑏} → (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))))
34	20, 33	syl6 35	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ 𝑋 ∈ 𝑉) ∧ 𝜓) → (∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → 𝑝 = 𝑞) → ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → (𝑝 = {𝑎, 𝑏} → (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏})))))
35	34	expimpd 456	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ 𝑋 ∈ 𝑉) → ((𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → 𝑝 = 𝑞)) → ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → (𝑝 = {𝑎, 𝑏} → (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏})))))
36	35	expimpd 456	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑋 ∈ 𝑉 ∧ (𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → 𝑝 = 𝑞))) → ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → (𝑝 = {𝑎, 𝑏} → (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏})))))
37	36	imp4c 426	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((((𝑋 ∈ 𝑉 ∧ (𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → 𝑝 = 𝑞))) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ 𝑝 = {𝑎, 𝑏}) → (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏})))
38	37	impcom 410	. . . . . . . . . . . 12 ⊢ (((((𝑋 ∈ 𝑉 ∧ (𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → 𝑝 = 𝑞))) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ 𝑝 = {𝑎, 𝑏}) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))
39	38	ralrimivva 3191	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ 𝑉 ∧ (𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → 𝑝 = 𝑞))) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ 𝑝 = {𝑎, 𝑏}) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))
40	11, 39	jca 514	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝑉 ∧ (𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → 𝑝 = 𝑞))) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ 𝑝 = {𝑎, 𝑏}) → (𝜒 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏})))
41	40	ex 415	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ (𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → 𝑝 = 𝑞))) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑝 = {𝑎, 𝑏} → (𝜒 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))))
42	41	reximdvva 3277	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → 𝑝 = 𝑞))) → (∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 𝑝 = {𝑎, 𝑏} → ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝜒 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))))
43	42	expcom 416	. . . . . . 7 ⊢ ((𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → 𝑝 = 𝑞)) → (𝑋 ∈ 𝑉 → (∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 𝑝 = {𝑎, 𝑏} → ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝜒 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏})))))
44	43	com13 88	. . . . . 6 ⊢ (∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 𝑝 = {𝑎, 𝑏} → (𝑋 ∈ 𝑉 → ((𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → 𝑝 = 𝑞)) → ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝜒 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏})))))
45	6, 44	syl 17	. . . . 5 ⊢ (𝑝 ∈ (Pairs‘𝑋) → (𝑋 ∈ 𝑉 → ((𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → 𝑝 = 𝑞)) → ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝜒 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏})))))
46	45	impcom 410	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑝 ∈ (Pairs‘𝑋)) → ((𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → 𝑝 = 𝑞)) → ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝜒 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))))
47	46	rexlimdva 3284	. . 3 ⊢ (𝑋 ∈ 𝑉 → (∃𝑝 ∈ (Pairs‘𝑋)(𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → 𝑝 = 𝑞)) → ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝜒 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))))
48		prelspr 43697	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑎, 𝑏} ∈ (Pairs‘𝑋))
49	48	adantr 483	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝜒 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))) → {𝑎, 𝑏} ∈ (Pairs‘𝑋))
50		simprl 769	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝜒 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))) → 𝜒)
51		nfsbc1v 3792	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥[𝑐 / 𝑥]𝜃
52		nfv 1915	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥{𝑐, 𝑦} = {𝑎, 𝑏}
53	51, 52	nfim 1897	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥([𝑐 / 𝑥]𝜃 → {𝑐, 𝑦} = {𝑎, 𝑏})
54		nfsbc1v 3792	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑦[𝑑 / 𝑦][𝑐 / 𝑥]𝜃
55		nfv 1915	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑦{𝑐, 𝑑} = {𝑎, 𝑏}
56	54, 55	nfim 1897	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑦([𝑑 / 𝑦][𝑐 / 𝑥]𝜃 → {𝑐, 𝑑} = {𝑎, 𝑏})
57		sbceq1a 3783	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑐 → (𝜃 ↔ [𝑐 / 𝑥]𝜃))
58		preq1 4669	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑐 → {𝑥, 𝑦} = {𝑐, 𝑦})
59	58	eqeq1d 2823	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑐 → ({𝑥, 𝑦} = {𝑎, 𝑏} ↔ {𝑐, 𝑦} = {𝑎, 𝑏}))
60	57, 59	imbi12d 347	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑐 → ((𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}) ↔ ([𝑐 / 𝑥]𝜃 → {𝑐, 𝑦} = {𝑎, 𝑏})))
61		sbceq1a 3783	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑑 → ([𝑐 / 𝑥]𝜃 ↔ [𝑑 / 𝑦][𝑐 / 𝑥]𝜃))
62		preq2 4670	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑑 → {𝑐, 𝑦} = {𝑐, 𝑑})
63	62	eqeq1d 2823	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑑 → ({𝑐, 𝑦} = {𝑎, 𝑏} ↔ {𝑐, 𝑑} = {𝑎, 𝑏}))
64	61, 63	imbi12d 347	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑑 → (([𝑐 / 𝑥]𝜃 → {𝑐, 𝑦} = {𝑎, 𝑏}) ↔ ([𝑑 / 𝑦][𝑐 / 𝑥]𝜃 → {𝑐, 𝑑} = {𝑎, 𝑏})))
65	53, 56, 60, 64	rspc2 3631	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ∈ 𝑋 ∧ 𝑑 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}) → ([𝑑 / 𝑦][𝑐 / 𝑥]𝜃 → {𝑐, 𝑑} = {𝑎, 𝑏})))
66	65	ad2antlr 725	. . . . . . . . . . . . . . 15 ⊢ ((((𝑋 ∈ 𝑉 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑐 ∈ 𝑋 ∧ 𝑑 ∈ 𝑋)) ∧ 𝜒) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}) → ([𝑑 / 𝑦][𝑐 / 𝑥]𝜃 → {𝑐, 𝑑} = {𝑎, 𝑏})))
67	22	sbcpr 43732	. . . . . . . . . . . . . . . . . 18 ⊢ ([{𝑐, 𝑑} / 𝑝]𝜓 ↔ [𝑑 / 𝑦][𝑐 / 𝑥]𝜃)
68		pm2.27 42	. . . . . . . . . . . . . . . . . 18 ⊢ ([𝑑 / 𝑦][𝑐 / 𝑥]𝜃 → (([𝑑 / 𝑦][𝑐 / 𝑥]𝜃 → {𝑐, 𝑑} = {𝑎, 𝑏}) → {𝑐, 𝑑} = {𝑎, 𝑏}))
69	67, 68	sylbi 219	. . . . . . . . . . . . . . . . 17 ⊢ ([{𝑐, 𝑑} / 𝑝]𝜓 → (([𝑑 / 𝑦][𝑐 / 𝑥]𝜃 → {𝑐, 𝑑} = {𝑎, 𝑏}) → {𝑐, 𝑑} = {𝑎, 𝑏}))
70		eqcom 2828	. . . . . . . . . . . . . . . . 17 ⊢ ({𝑎, 𝑏} = {𝑐, 𝑑} ↔ {𝑐, 𝑑} = {𝑎, 𝑏})
71	69, 70	syl6ibr 254	. . . . . . . . . . . . . . . 16 ⊢ ([{𝑐, 𝑑} / 𝑝]𝜓 → (([𝑑 / 𝑦][𝑐 / 𝑥]𝜃 → {𝑐, 𝑑} = {𝑎, 𝑏}) → {𝑎, 𝑏} = {𝑐, 𝑑}))
72	71	com12 32	. . . . . . . . . . . . . . 15 ⊢ (([𝑑 / 𝑦][𝑐 / 𝑥]𝜃 → {𝑐, 𝑑} = {𝑎, 𝑏}) → ([{𝑐, 𝑑} / 𝑝]𝜓 → {𝑎, 𝑏} = {𝑐, 𝑑}))
73	66, 72	syl6 35	. . . . . . . . . . . . . 14 ⊢ ((((𝑋 ∈ 𝑉 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑐 ∈ 𝑋 ∧ 𝑑 ∈ 𝑋)) ∧ 𝜒) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}) → ([{𝑐, 𝑑} / 𝑝]𝜓 → {𝑎, 𝑏} = {𝑐, 𝑑})))
74	73	expimpd 456	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ 𝑉 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑐 ∈ 𝑋 ∧ 𝑑 ∈ 𝑋)) → ((𝜒 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏})) → ([{𝑐, 𝑑} / 𝑝]𝜓 → {𝑎, 𝑏} = {𝑐, 𝑑})))
75	74	expcom 416	. . . . . . . . . . . 12 ⊢ ((𝑐 ∈ 𝑋 ∧ 𝑑 ∈ 𝑋) → ((𝑋 ∈ 𝑉 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝜒 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏})) → ([{𝑐, 𝑑} / 𝑝]𝜓 → {𝑎, 𝑏} = {𝑐, 𝑑}))))
76	75	impd 413	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ 𝑋 ∧ 𝑑 ∈ 𝑋) → (((𝑋 ∈ 𝑉 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝜒 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))) → ([{𝑐, 𝑑} / 𝑝]𝜓 → {𝑎, 𝑏} = {𝑐, 𝑑})))
77	76	impcom 410	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝑉 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝜒 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))) ∧ (𝑐 ∈ 𝑋 ∧ 𝑑 ∈ 𝑋)) → ([{𝑐, 𝑑} / 𝑝]𝜓 → {𝑎, 𝑏} = {𝑐, 𝑑}))
78		dfsbcq 3774	. . . . . . . . . . 11 ⊢ (𝑞 = {𝑐, 𝑑} → ([𝑞 / 𝑝]𝜓 ↔ [{𝑐, 𝑑} / 𝑝]𝜓))
79		eqeq2 2833	. . . . . . . . . . 11 ⊢ (𝑞 = {𝑐, 𝑑} → ({𝑎, 𝑏} = 𝑞 ↔ {𝑎, 𝑏} = {𝑐, 𝑑}))
80	78, 79	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑞 = {𝑐, 𝑑} → (([𝑞 / 𝑝]𝜓 → {𝑎, 𝑏} = 𝑞) ↔ ([{𝑐, 𝑑} / 𝑝]𝜓 → {𝑎, 𝑏} = {𝑐, 𝑑})))
81	77, 80	syl5ibrcom 249	. . . . . . . . 9 ⊢ ((((𝑋 ∈ 𝑉 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝜒 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))) ∧ (𝑐 ∈ 𝑋 ∧ 𝑑 ∈ 𝑋)) → (𝑞 = {𝑐, 𝑑} → ([𝑞 / 𝑝]𝜓 → {𝑎, 𝑏} = 𝑞)))
82	81	rexlimdvva 3294	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝜒 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))) → (∃𝑐 ∈ 𝑋 ∃𝑑 ∈ 𝑋 𝑞 = {𝑐, 𝑑} → ([𝑞 / 𝑝]𝜓 → {𝑎, 𝑏} = 𝑞)))
83		sprel 43695	. . . . . . . 8 ⊢ (𝑞 ∈ (Pairs‘𝑋) → ∃𝑐 ∈ 𝑋 ∃𝑑 ∈ 𝑋 𝑞 = {𝑐, 𝑑})
84	82, 83	impel 508	. . . . . . 7 ⊢ ((((𝑋 ∈ 𝑉 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝜒 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))) ∧ 𝑞 ∈ (Pairs‘𝑋)) → ([𝑞 / 𝑝]𝜓 → {𝑎, 𝑏} = 𝑞))
85	84	ralrimiva 3182	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝜒 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))) → ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → {𝑎, 𝑏} = 𝑞))
86		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑝𝜒
87		nfcv 2977	. . . . . . . . 9 ⊢ Ⅎ𝑝(Pairs‘𝑋)
88		nfv 1915	. . . . . . . . . 10 ⊢ Ⅎ𝑝{𝑎, 𝑏} = 𝑞
89	1, 88	nfim 1897	. . . . . . . . 9 ⊢ Ⅎ𝑝([𝑞 / 𝑝]𝜓 → {𝑎, 𝑏} = 𝑞)
90	87, 89	nfralw 3225	. . . . . . . 8 ⊢ Ⅎ𝑝∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → {𝑎, 𝑏} = 𝑞)
91	86, 90	nfan 1900	. . . . . . 7 ⊢ Ⅎ𝑝(𝜒 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → {𝑎, 𝑏} = 𝑞))
92		eqeq1 2825	. . . . . . . . . 10 ⊢ (𝑝 = {𝑎, 𝑏} → (𝑝 = 𝑞 ↔ {𝑎, 𝑏} = 𝑞))
93	92	imbi2d 343	. . . . . . . . 9 ⊢ (𝑝 = {𝑎, 𝑏} → (([𝑞 / 𝑝]𝜓 → 𝑝 = 𝑞) ↔ ([𝑞 / 𝑝]𝜓 → {𝑎, 𝑏} = 𝑞)))
94	93	ralbidv 3197	. . . . . . . 8 ⊢ (𝑝 = {𝑎, 𝑏} → (∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → 𝑝 = 𝑞) ↔ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → {𝑎, 𝑏} = 𝑞)))
95	7, 94	anbi12d 632	. . . . . . 7 ⊢ (𝑝 = {𝑎, 𝑏} → ((𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → 𝑝 = 𝑞)) ↔ (𝜒 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → {𝑎, 𝑏} = 𝑞))))
96	91, 95	rspce 3612	. . . . . 6 ⊢ (({𝑎, 𝑏} ∈ (Pairs‘𝑋) ∧ (𝜒 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → {𝑎, 𝑏} = 𝑞))) → ∃𝑝 ∈ (Pairs‘𝑋)(𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → 𝑝 = 𝑞)))
97	49, 50, 85, 96	syl12anc 834	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝜒 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))) → ∃𝑝 ∈ (Pairs‘𝑋)(𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → 𝑝 = 𝑞)))
98	97	ex 415	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((𝜒 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏})) → ∃𝑝 ∈ (Pairs‘𝑋)(𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → 𝑝 = 𝑞))))
99	98	rexlimdvva 3294	. . 3 ⊢ (𝑋 ∈ 𝑉 → (∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝜒 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏})) → ∃𝑝 ∈ (Pairs‘𝑋)(𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → 𝑝 = 𝑞))))
100	47, 99	impbid 214	. 2 ⊢ (𝑋 ∈ 𝑉 → (∃𝑝 ∈ (Pairs‘𝑋)(𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → 𝑝 = 𝑞)) ↔ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝜒 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))))
101	5, 100	syl5bb 285	1 ⊢ (𝑋 ∈ 𝑉 → (∃!𝑝 ∈ (Pairs‘𝑋)𝜓 ↔ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝜒 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3138  ∃wrex 3139  ∃!wreu 3140  [wsbc 3772  {cpr 4569  ‘cfv 6355  Pairscspr 43688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-spr 43689

This theorem is referenced by:  reuprpr  43734  reuopreuprim  43737