Theorem reupr 43180

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 3727	. . 3 ⊢ Ⅎ𝑝[𝑞 / 𝑝]𝜓
2		nfsbc1v 3727	. . 3 ⊢ Ⅎ𝑝[𝑤 / 𝑝]𝜓
3		sbceq1a 3718	. . 3 ⊢ (𝑝 = 𝑤 → (𝜓 ↔ [𝑤 / 𝑝]𝜓))
4		dfsbcq 3709	. . 3 ⊢ (𝑤 = 𝑞 → ([𝑤 / 𝑝]𝜓 ↔ [𝑞 / 𝑝]𝜓))
5	1, 2, 3, 4	reu8nf 3790	. 2 ⊢ (∃!𝑝 ∈ (Pairs‘𝑋)𝜓 ↔ ∃𝑝 ∈ (Pairs‘𝑋)(𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → 𝑝 = 𝑞)))
6		sprel 43142	. . . . . 6 ⊢ (𝑝 ∈ (Pairs‘𝑋) → ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 𝑝 = {𝑎, 𝑏})
7		reupr.a	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = {𝑎, 𝑏} → (𝜓 ↔ 𝜒))
8	7	biimpcd 250	. . . . . . . . . . . . . 14 ⊢ (𝜓 → (𝑝 = {𝑎, 𝑏} → 𝜒))
9	8	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → 𝑝 = 𝑞)) → (𝑝 = {𝑎, 𝑏} → 𝜒))
10	9	ad2antlr 723	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ 𝑉 ∧ (𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → 𝑝 = 𝑞))) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑝 = {𝑎, 𝑏} → 𝜒))
11	10	imp 407	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ 𝑉 ∧ (𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → 𝑝 = 𝑞))) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ 𝑝 = {𝑎, 𝑏}) → 𝜒)
12		pm3.22 460	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ 𝑋 ∈ 𝑉) → (𝑋 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)))
13	12	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ 𝑋 ∈ 𝑉) ∧ 𝜓) → (𝑋 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)))
14		prelspr 43144	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → {𝑥, 𝑦} ∈ (Pairs‘𝑋))
15		dfsbcq 3709	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑞 = {𝑥, 𝑦} → ([𝑞 / 𝑝]𝜓 ↔ [{𝑥, 𝑦} / 𝑝]𝜓))
16		eqeq2 2805	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑞 = {𝑥, 𝑦} → (𝑝 = 𝑞 ↔ 𝑝 = {𝑥, 𝑦}))
17	15, 16	imbi12d 346	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑞 = {𝑥, 𝑦} → (([𝑞 / 𝑝]𝜓 → 𝑝 = 𝑞) ↔ ([{𝑥, 𝑦} / 𝑝]𝜓 → 𝑝 = {𝑥, 𝑦})))
18	17	adantl 482	. . . . . . . . . . . . . . . . . . 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 3560	. . . . . . . . . . . . . . . . . 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 5225	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ {𝑥, 𝑦} ∈ V
22		reupr.x	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑝 = {𝑥, 𝑦} → (𝜓 ↔ 𝜃))
23	21, 22	sbcie 3742	. . . . . . . . . . . . . . . . . . . . . 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 236	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜃 → (([{𝑥, 𝑦} / 𝑝]𝜓 → 𝑝 = {𝑥, 𝑦}) → 𝑝 = {𝑥, 𝑦}))
26		eqcom 2801	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({𝑥, 𝑦} = 𝑝 ↔ 𝑝 = {𝑥, 𝑦})
27	25, 26	syl6ibr 253	. . . . . . . . . . . . . . . . . . . 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 2805	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({𝑎, 𝑏} = 𝑝 → ({𝑥, 𝑦} = {𝑎, 𝑏} ↔ {𝑥, 𝑦} = 𝑝))
30	29	eqcoms 2802	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑝 = {𝑎, 𝑏} → ({𝑥, 𝑦} = {𝑎, 𝑏} ↔ {𝑥, 𝑦} = 𝑝))
31	30	imbi2d 342	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 = {𝑎, 𝑏} → ((𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}) ↔ (𝜃 → {𝑥, 𝑦} = 𝑝)))
32	28, 31	syl5ibrcom 248	. . . . . . . . . . . . . . . . . 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 454	. . . . . . . . . . . . . . 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 454	. . . . . . . . . . . . . 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 424	. . . . . . . . . . . . 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 408	. . . . . . . . . . . 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 3157	. . . . . . . . . . 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 512	. . . . . . . . . 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 413	. . . . . . . . 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 3239	. . . . . . . 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 414	. . . . . . 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 408	. . . 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 3246	. . 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 43144	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑎, 𝑏} ∈ (Pairs‘𝑋))
49	48	adantr 481	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝜒 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))) → {𝑎, 𝑏} ∈ (Pairs‘𝑋))
50		simprl 767	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝜒 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))) → 𝜒)
51		nfsbc1v 3727	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥[𝑐 / 𝑥]𝜃
52		nfv 1893	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥{𝑐, 𝑦} = {𝑎, 𝑏}
53	51, 52	nfim 1879	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥([𝑐 / 𝑥]𝜃 → {𝑐, 𝑦} = {𝑎, 𝑏})
54		nfsbc1v 3727	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑦[𝑑 / 𝑦][𝑐 / 𝑥]𝜃
55		nfv 1893	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑦{𝑐, 𝑑} = {𝑎, 𝑏}
56	54, 55	nfim 1879	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑦([𝑑 / 𝑦][𝑐 / 𝑥]𝜃 → {𝑐, 𝑑} = {𝑎, 𝑏})
57		sbceq1a 3718	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑐 → (𝜃 ↔ [𝑐 / 𝑥]𝜃))
58		preq1 4578	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑐 → {𝑥, 𝑦} = {𝑐, 𝑦})
59	58	eqeq1d 2796	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑐 → ({𝑥, 𝑦} = {𝑎, 𝑏} ↔ {𝑐, 𝑦} = {𝑎, 𝑏}))
60	57, 59	imbi12d 346	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑐 → ((𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}) ↔ ([𝑐 / 𝑥]𝜃 → {𝑐, 𝑦} = {𝑎, 𝑏})))
61		sbceq1a 3718	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑑 → ([𝑐 / 𝑥]𝜃 ↔ [𝑑 / 𝑦][𝑐 / 𝑥]𝜃))
62		preq2 4579	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑑 → {𝑐, 𝑦} = {𝑐, 𝑑})
63	62	eqeq1d 2796	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑑 → ({𝑐, 𝑦} = {𝑎, 𝑏} ↔ {𝑐, 𝑑} = {𝑎, 𝑏}))
64	61, 63	imbi12d 346	. . . . . . . . . . . . . . . . 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 3568	. . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . 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 43179	. . . . . . . . . . . . . . . . . 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 218	. . . . . . . . . . . . . . . . 17 ⊢ ([{𝑐, 𝑑} / 𝑝]𝜓 → (([𝑑 / 𝑦][𝑐 / 𝑥]𝜃 → {𝑐, 𝑑} = {𝑎, 𝑏}) → {𝑐, 𝑑} = {𝑎, 𝑏}))
70		eqcom 2801	. . . . . . . . . . . . . . . . 17 ⊢ ({𝑎, 𝑏} = {𝑐, 𝑑} ↔ {𝑐, 𝑑} = {𝑎, 𝑏})
71	69, 70	syl6ibr 253	. . . . . . . . . . . . . . . 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 454	. . . . . . . . . . . . 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 414	. . . . . . . . . . . 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 411	. . . . . . . . . . 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 408	. . . . . . . . . 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 3709	. . . . . . . . . . 11 ⊢ (𝑞 = {𝑐, 𝑑} → ([𝑞 / 𝑝]𝜓 ↔ [{𝑐, 𝑑} / 𝑝]𝜓))
79		eqeq2 2805	. . . . . . . . . . 11 ⊢ (𝑞 = {𝑐, 𝑑} → ({𝑎, 𝑏} = 𝑞 ↔ {𝑎, 𝑏} = {𝑐, 𝑑}))
80	78, 79	imbi12d 346	. . . . . . . . . 10 ⊢ (𝑞 = {𝑐, 𝑑} → (([𝑞 / 𝑝]𝜓 → {𝑎, 𝑏} = 𝑞) ↔ ([{𝑐, 𝑑} / 𝑝]𝜓 → {𝑎, 𝑏} = {𝑐, 𝑑})))
81	77, 80	syl5ibrcom 248	. . . . . . . . 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 3256	. . . . . . . 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 43142	. . . . . . . 8 ⊢ (𝑞 ∈ (Pairs‘𝑋) → ∃𝑐 ∈ 𝑋 ∃𝑑 ∈ 𝑋 𝑞 = {𝑐, 𝑑})
84	82, 83	impel 506	. . . . . . 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 3148	. . . . . 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 1893	. . . . . . . 8 ⊢ Ⅎ𝑝𝜒
87		nfcv 2948	. . . . . . . . 9 ⊢ Ⅎ𝑝(Pairs‘𝑋)
88		nfv 1893	. . . . . . . . . 10 ⊢ Ⅎ𝑝{𝑎, 𝑏} = 𝑞
89	1, 88	nfim 1879	. . . . . . . . 9 ⊢ Ⅎ𝑝([𝑞 / 𝑝]𝜓 → {𝑎, 𝑏} = 𝑞)
90	87, 89	nfral 3190	. . . . . . . 8 ⊢ Ⅎ𝑝∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → {𝑎, 𝑏} = 𝑞)
91	86, 90	nfan 1882	. . . . . . 7 ⊢ Ⅎ𝑝(𝜒 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → {𝑎, 𝑏} = 𝑞))
92		eqeq1 2798	. . . . . . . . . 10 ⊢ (𝑝 = {𝑎, 𝑏} → (𝑝 = 𝑞 ↔ {𝑎, 𝑏} = 𝑞))
93	92	imbi2d 342	. . . . . . . . 9 ⊢ (𝑝 = {𝑎, 𝑏} → (([𝑞 / 𝑝]𝜓 → 𝑝 = 𝑞) ↔ ([𝑞 / 𝑝]𝜓 → {𝑎, 𝑏} = 𝑞)))
94	93	ralbidv 3163	. . . . . . . 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 630	. . . . . . 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 3552	. . . . . 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 833	. . . . 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 413	. . . 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 3256	. . 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 213	. 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 284	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 207   ∧ wa 396   = wceq 1522   ∈ wcel 2080  ∀wral 3104  ∃wrex 3105  ∃!wreu 3106  [wsbc 3707  {cpr 4476  ‘cfv 6228  Pairscspr 43135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-rep 5084  ax-sep 5097  ax-nul 5104  ax-pow 5160  ax-pr 5224  ax-un 7322

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ne 2984  df-ral 3109  df-rex 3110  df-reu 3111  df-rab 3113  df-v 3438  df-sbc 3708  df-csb 3814  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-nul 4214  df-if 4384  df-pw 4457  df-sn 4475  df-pr 4477  df-op 4481  df-uni 4748  df-iun 4829  df-br 4965  df-opab 5027  df-mpt 5044  df-id 5351  df-xp 5452  df-rel 5453  df-cnv 5454  df-co 5455  df-dm 5456  df-rn 5457  df-res 5458  df-ima 5459  df-iota 6192  df-fun 6230  df-fn 6231  df-f 6232  df-f1 6233  df-fo 6234  df-f1o 6235  df-fv 6236  df-spr 43136

This theorem is referenced by:  reuprpr  43181  reuopreuprim  43184