Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  reupr Structured version   Visualization version   GIF version

Theorem reupr 44054
 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} → (𝜓𝜒))
reupr.x (𝑝 = {𝑥, 𝑦} → (𝜓𝜃))
Assertion
Ref Expression
reupr (𝑋𝑉 → (∃!𝑝 ∈ (Pairs‘𝑋)𝜓 ↔ ∃𝑎𝑋𝑏𝑋 (𝜒 ∧ ∀𝑥𝑋𝑦𝑋 (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.
StepHypRef Expression
1 nfsbc1v 3740 . . 3 𝑝[𝑞 / 𝑝]𝜓
2 nfsbc1v 3740 . . 3 𝑝[𝑤 / 𝑝]𝜓
3 sbceq1a 3731 . . 3 (𝑝 = 𝑤 → (𝜓[𝑤 / 𝑝]𝜓))
4 dfsbcq 3722 . . 3 (𝑤 = 𝑞 → ([𝑤 / 𝑝]𝜓[𝑞 / 𝑝]𝜓))
51, 2, 3, 4reu8nf 3806 . 2 (∃!𝑝 ∈ (Pairs‘𝑋)𝜓 ↔ ∃𝑝 ∈ (Pairs‘𝑋)(𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓𝑝 = 𝑞)))
6 sprel 44016 . . . . . 6 (𝑝 ∈ (Pairs‘𝑋) → ∃𝑎𝑋𝑏𝑋 𝑝 = {𝑎, 𝑏})
7 reupr.a . . . . . . . . . . . . . . 15 (𝑝 = {𝑎, 𝑏} → (𝜓𝜒))
87biimpcd 252 . . . . . . . . . . . . . 14 (𝜓 → (𝑝 = {𝑎, 𝑏} → 𝜒))
98adantr 484 . . . . . . . . . . . . 13 ((𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓𝑝 = 𝑞)) → (𝑝 = {𝑎, 𝑏} → 𝜒))
109ad2antlr 726 . . . . . . . . . . . 12 (((𝑋𝑉 ∧ (𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓𝑝 = 𝑞))) ∧ (𝑎𝑋𝑏𝑋)) → (𝑝 = {𝑎, 𝑏} → 𝜒))
1110imp 410 . . . . . . . . . . 11 ((((𝑋𝑉 ∧ (𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓𝑝 = 𝑞))) ∧ (𝑎𝑋𝑏𝑋)) ∧ 𝑝 = {𝑎, 𝑏}) → 𝜒)
12 pm3.22 463 . . . . . . . . . . . . . . . . . . 19 (((𝑥𝑋𝑦𝑋) ∧ 𝑋𝑉) → (𝑋𝑉 ∧ (𝑥𝑋𝑦𝑋)))
1312adantr 484 . . . . . . . . . . . . . . . . . 18 ((((𝑥𝑋𝑦𝑋) ∧ 𝑋𝑉) ∧ 𝜓) → (𝑋𝑉 ∧ (𝑥𝑋𝑦𝑋)))
14 prelspr 44018 . . . . . . . . . . . . . . . . . . 19 ((𝑋𝑉 ∧ (𝑥𝑋𝑦𝑋)) → {𝑥, 𝑦} ∈ (Pairs‘𝑋))
15 dfsbcq 3722 . . . . . . . . . . . . . . . . . . . . 21 (𝑞 = {𝑥, 𝑦} → ([𝑞 / 𝑝]𝜓[{𝑥, 𝑦} / 𝑝]𝜓))
16 eqeq2 2810 . . . . . . . . . . . . . . . . . . . . 21 (𝑞 = {𝑥, 𝑦} → (𝑝 = 𝑞𝑝 = {𝑥, 𝑦}))
1715, 16imbi12d 348 . . . . . . . . . . . . . . . . . . . 20 (𝑞 = {𝑥, 𝑦} → (([𝑞 / 𝑝]𝜓𝑝 = 𝑞) ↔ ([{𝑥, 𝑦} / 𝑝]𝜓𝑝 = {𝑥, 𝑦})))
1817adantl 485 . . . . . . . . . . . . . . . . . . 19 (((𝑋𝑉 ∧ (𝑥𝑋𝑦𝑋)) ∧ 𝑞 = {𝑥, 𝑦}) → (([𝑞 / 𝑝]𝜓𝑝 = 𝑞) ↔ ([{𝑥, 𝑦} / 𝑝]𝜓𝑝 = {𝑥, 𝑦})))
1914, 18rspcdv 3563 . . . . . . . . . . . . . . . . . 18 ((𝑋𝑉 ∧ (𝑥𝑋𝑦𝑋)) → (∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓𝑝 = 𝑞) → ([{𝑥, 𝑦} / 𝑝]𝜓𝑝 = {𝑥, 𝑦})))
2013, 19syl 17 . . . . . . . . . . . . . . . . 17 ((((𝑥𝑋𝑦𝑋) ∧ 𝑋𝑉) ∧ 𝜓) → (∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓𝑝 = 𝑞) → ([{𝑥, 𝑦} / 𝑝]𝜓𝑝 = {𝑥, 𝑦})))
21 zfpair2 5296 . . . . . . . . . . . . . . . . . . . . . . 23 {𝑥, 𝑦} ∈ V
22 reupr.x . . . . . . . . . . . . . . . . . . . . . . 23 (𝑝 = {𝑥, 𝑦} → (𝜓𝜃))
2321, 22sbcie 3760 . . . . . . . . . . . . . . . . . . . . . 22 ([{𝑥, 𝑦} / 𝑝]𝜓𝜃)
24 pm2.27 42 . . . . . . . . . . . . . . . . . . . . . 22 ([{𝑥, 𝑦} / 𝑝]𝜓 → (([{𝑥, 𝑦} / 𝑝]𝜓𝑝 = {𝑥, 𝑦}) → 𝑝 = {𝑥, 𝑦}))
2523, 24sylbir 238 . . . . . . . . . . . . . . . . . . . . 21 (𝜃 → (([{𝑥, 𝑦} / 𝑝]𝜓𝑝 = {𝑥, 𝑦}) → 𝑝 = {𝑥, 𝑦}))
26 eqcom 2805 . . . . . . . . . . . . . . . . . . . . 21 ({𝑥, 𝑦} = 𝑝𝑝 = {𝑥, 𝑦})
2725, 26syl6ibr 255 . . . . . . . . . . . . . . . . . . . 20 (𝜃 → (([{𝑥, 𝑦} / 𝑝]𝜓𝑝 = {𝑥, 𝑦}) → {𝑥, 𝑦} = 𝑝))
2827com12 32 . . . . . . . . . . . . . . . . . . 19 (([{𝑥, 𝑦} / 𝑝]𝜓𝑝 = {𝑥, 𝑦}) → (𝜃 → {𝑥, 𝑦} = 𝑝))
29 eqeq2 2810 . . . . . . . . . . . . . . . . . . . . 21 ({𝑎, 𝑏} = 𝑝 → ({𝑥, 𝑦} = {𝑎, 𝑏} ↔ {𝑥, 𝑦} = 𝑝))
3029eqcoms 2806 . . . . . . . . . . . . . . . . . . . 20 (𝑝 = {𝑎, 𝑏} → ({𝑥, 𝑦} = {𝑎, 𝑏} ↔ {𝑥, 𝑦} = 𝑝))
3130imbi2d 344 . . . . . . . . . . . . . . . . . . 19 (𝑝 = {𝑎, 𝑏} → ((𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}) ↔ (𝜃 → {𝑥, 𝑦} = 𝑝)))
3228, 31syl5ibrcom 250 . . . . . . . . . . . . . . . . . 18 (([{𝑥, 𝑦} / 𝑝]𝜓𝑝 = {𝑥, 𝑦}) → (𝑝 = {𝑎, 𝑏} → (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏})))
3332a1d 25 . . . . . . . . . . . . . . . . 17 (([{𝑥, 𝑦} / 𝑝]𝜓𝑝 = {𝑥, 𝑦}) → ((𝑎𝑋𝑏𝑋) → (𝑝 = {𝑎, 𝑏} → (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))))
3420, 33syl6 35 . . . . . . . . . . . . . . . 16 ((((𝑥𝑋𝑦𝑋) ∧ 𝑋𝑉) ∧ 𝜓) → (∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓𝑝 = 𝑞) → ((𝑎𝑋𝑏𝑋) → (𝑝 = {𝑎, 𝑏} → (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏})))))
3534expimpd 457 . . . . . . . . . . . . . . 15 (((𝑥𝑋𝑦𝑋) ∧ 𝑋𝑉) → ((𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓𝑝 = 𝑞)) → ((𝑎𝑋𝑏𝑋) → (𝑝 = {𝑎, 𝑏} → (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏})))))
3635expimpd 457 . . . . . . . . . . . . . 14 ((𝑥𝑋𝑦𝑋) → ((𝑋𝑉 ∧ (𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓𝑝 = 𝑞))) → ((𝑎𝑋𝑏𝑋) → (𝑝 = {𝑎, 𝑏} → (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏})))))
3736imp4c 427 . . . . . . . . . . . . 13 ((𝑥𝑋𝑦𝑋) → ((((𝑋𝑉 ∧ (𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓𝑝 = 𝑞))) ∧ (𝑎𝑋𝑏𝑋)) ∧ 𝑝 = {𝑎, 𝑏}) → (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏})))
3837impcom 411 . . . . . . . . . . . 12 (((((𝑋𝑉 ∧ (𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓𝑝 = 𝑞))) ∧ (𝑎𝑋𝑏𝑋)) ∧ 𝑝 = {𝑎, 𝑏}) ∧ (𝑥𝑋𝑦𝑋)) → (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))
3938ralrimivva 3156 . . . . . . . . . . 11 ((((𝑋𝑉 ∧ (𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓𝑝 = 𝑞))) ∧ (𝑎𝑋𝑏𝑋)) ∧ 𝑝 = {𝑎, 𝑏}) → ∀𝑥𝑋𝑦𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))
4011, 39jca 515 . . . . . . . . . 10 ((((𝑋𝑉 ∧ (𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓𝑝 = 𝑞))) ∧ (𝑎𝑋𝑏𝑋)) ∧ 𝑝 = {𝑎, 𝑏}) → (𝜒 ∧ ∀𝑥𝑋𝑦𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏})))
4140ex 416 . . . . . . . . 9 (((𝑋𝑉 ∧ (𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓𝑝 = 𝑞))) ∧ (𝑎𝑋𝑏𝑋)) → (𝑝 = {𝑎, 𝑏} → (𝜒 ∧ ∀𝑥𝑋𝑦𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))))
4241reximdvva 3236 . . . . . . . 8 ((𝑋𝑉 ∧ (𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓𝑝 = 𝑞))) → (∃𝑎𝑋𝑏𝑋 𝑝 = {𝑎, 𝑏} → ∃𝑎𝑋𝑏𝑋 (𝜒 ∧ ∀𝑥𝑋𝑦𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))))
4342expcom 417 . . . . . . 7 ((𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓𝑝 = 𝑞)) → (𝑋𝑉 → (∃𝑎𝑋𝑏𝑋 𝑝 = {𝑎, 𝑏} → ∃𝑎𝑋𝑏𝑋 (𝜒 ∧ ∀𝑥𝑋𝑦𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏})))))
4443com13 88 . . . . . 6 (∃𝑎𝑋𝑏𝑋 𝑝 = {𝑎, 𝑏} → (𝑋𝑉 → ((𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓𝑝 = 𝑞)) → ∃𝑎𝑋𝑏𝑋 (𝜒 ∧ ∀𝑥𝑋𝑦𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏})))))
456, 44syl 17 . . . . 5 (𝑝 ∈ (Pairs‘𝑋) → (𝑋𝑉 → ((𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓𝑝 = 𝑞)) → ∃𝑎𝑋𝑏𝑋 (𝜒 ∧ ∀𝑥𝑋𝑦𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏})))))
4645impcom 411 . . . 4 ((𝑋𝑉𝑝 ∈ (Pairs‘𝑋)) → ((𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓𝑝 = 𝑞)) → ∃𝑎𝑋𝑏𝑋 (𝜒 ∧ ∀𝑥𝑋𝑦𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))))
4746rexlimdva 3243 . . 3 (𝑋𝑉 → (∃𝑝 ∈ (Pairs‘𝑋)(𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓𝑝 = 𝑞)) → ∃𝑎𝑋𝑏𝑋 (𝜒 ∧ ∀𝑥𝑋𝑦𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))))
48 prelspr 44018 . . . . . . 7 ((𝑋𝑉 ∧ (𝑎𝑋𝑏𝑋)) → {𝑎, 𝑏} ∈ (Pairs‘𝑋))
4948adantr 484 . . . . . 6 (((𝑋𝑉 ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝜒 ∧ ∀𝑥𝑋𝑦𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))) → {𝑎, 𝑏} ∈ (Pairs‘𝑋))
50 simprl 770 . . . . . 6 (((𝑋𝑉 ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝜒 ∧ ∀𝑥𝑋𝑦𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))) → 𝜒)
51 nfsbc1v 3740 . . . . . . . . . . . . . . . . . 18 𝑥[𝑐 / 𝑥]𝜃
52 nfv 1915 . . . . . . . . . . . . . . . . . 18 𝑥{𝑐, 𝑦} = {𝑎, 𝑏}
5351, 52nfim 1897 . . . . . . . . . . . . . . . . 17 𝑥([𝑐 / 𝑥]𝜃 → {𝑐, 𝑦} = {𝑎, 𝑏})
54 nfsbc1v 3740 . . . . . . . . . . . . . . . . . 18 𝑦[𝑑 / 𝑦][𝑐 / 𝑥]𝜃
55 nfv 1915 . . . . . . . . . . . . . . . . . 18 𝑦{𝑐, 𝑑} = {𝑎, 𝑏}
5654, 55nfim 1897 . . . . . . . . . . . . . . . . 17 𝑦([𝑑 / 𝑦][𝑐 / 𝑥]𝜃 → {𝑐, 𝑑} = {𝑎, 𝑏})
57 sbceq1a 3731 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑐 → (𝜃[𝑐 / 𝑥]𝜃))
58 preq1 4629 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑐 → {𝑥, 𝑦} = {𝑐, 𝑦})
5958eqeq1d 2800 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑐 → ({𝑥, 𝑦} = {𝑎, 𝑏} ↔ {𝑐, 𝑦} = {𝑎, 𝑏}))
6057, 59imbi12d 348 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑐 → ((𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}) ↔ ([𝑐 / 𝑥]𝜃 → {𝑐, 𝑦} = {𝑎, 𝑏})))
61 sbceq1a 3731 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑑 → ([𝑐 / 𝑥]𝜃[𝑑 / 𝑦][𝑐 / 𝑥]𝜃))
62 preq2 4630 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑑 → {𝑐, 𝑦} = {𝑐, 𝑑})
6362eqeq1d 2800 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑑 → ({𝑐, 𝑦} = {𝑎, 𝑏} ↔ {𝑐, 𝑑} = {𝑎, 𝑏}))
6461, 63imbi12d 348 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑑 → (([𝑐 / 𝑥]𝜃 → {𝑐, 𝑦} = {𝑎, 𝑏}) ↔ ([𝑑 / 𝑦][𝑐 / 𝑥]𝜃 → {𝑐, 𝑑} = {𝑎, 𝑏})))
6553, 56, 60, 64rspc2 3579 . . . . . . . . . . . . . . . 16 ((𝑐𝑋𝑑𝑋) → (∀𝑥𝑋𝑦𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}) → ([𝑑 / 𝑦][𝑐 / 𝑥]𝜃 → {𝑐, 𝑑} = {𝑎, 𝑏})))
6665ad2antlr 726 . . . . . . . . . . . . . . 15 ((((𝑋𝑉 ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝑋𝑑𝑋)) ∧ 𝜒) → (∀𝑥𝑋𝑦𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}) → ([𝑑 / 𝑦][𝑐 / 𝑥]𝜃 → {𝑐, 𝑑} = {𝑎, 𝑏})))
6722sbcpr 44053 . . . . . . . . . . . . . . . . . 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}))
6967, 68sylbi 220 . . . . . . . . . . . . . . . . 17 ([{𝑐, 𝑑} / 𝑝]𝜓 → (([𝑑 / 𝑦][𝑐 / 𝑥]𝜃 → {𝑐, 𝑑} = {𝑎, 𝑏}) → {𝑐, 𝑑} = {𝑎, 𝑏}))
70 eqcom 2805 . . . . . . . . . . . . . . . . 17 ({𝑎, 𝑏} = {𝑐, 𝑑} ↔ {𝑐, 𝑑} = {𝑎, 𝑏})
7169, 70syl6ibr 255 . . . . . . . . . . . . . . . 16 ([{𝑐, 𝑑} / 𝑝]𝜓 → (([𝑑 / 𝑦][𝑐 / 𝑥]𝜃 → {𝑐, 𝑑} = {𝑎, 𝑏}) → {𝑎, 𝑏} = {𝑐, 𝑑}))
7271com12 32 . . . . . . . . . . . . . . 15 (([𝑑 / 𝑦][𝑐 / 𝑥]𝜃 → {𝑐, 𝑑} = {𝑎, 𝑏}) → ([{𝑐, 𝑑} / 𝑝]𝜓 → {𝑎, 𝑏} = {𝑐, 𝑑}))
7366, 72syl6 35 . . . . . . . . . . . . . 14 ((((𝑋𝑉 ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝑋𝑑𝑋)) ∧ 𝜒) → (∀𝑥𝑋𝑦𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}) → ([{𝑐, 𝑑} / 𝑝]𝜓 → {𝑎, 𝑏} = {𝑐, 𝑑})))
7473expimpd 457 . . . . . . . . . . . . 13 (((𝑋𝑉 ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝑋𝑑𝑋)) → ((𝜒 ∧ ∀𝑥𝑋𝑦𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏})) → ([{𝑐, 𝑑} / 𝑝]𝜓 → {𝑎, 𝑏} = {𝑐, 𝑑})))
7574expcom 417 . . . . . . . . . . . 12 ((𝑐𝑋𝑑𝑋) → ((𝑋𝑉 ∧ (𝑎𝑋𝑏𝑋)) → ((𝜒 ∧ ∀𝑥𝑋𝑦𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏})) → ([{𝑐, 𝑑} / 𝑝]𝜓 → {𝑎, 𝑏} = {𝑐, 𝑑}))))
7675impd 414 . . . . . . . . . . 11 ((𝑐𝑋𝑑𝑋) → (((𝑋𝑉 ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝜒 ∧ ∀𝑥𝑋𝑦𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))) → ([{𝑐, 𝑑} / 𝑝]𝜓 → {𝑎, 𝑏} = {𝑐, 𝑑})))
7776impcom 411 . . . . . . . . . 10 ((((𝑋𝑉 ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝜒 ∧ ∀𝑥𝑋𝑦𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))) ∧ (𝑐𝑋𝑑𝑋)) → ([{𝑐, 𝑑} / 𝑝]𝜓 → {𝑎, 𝑏} = {𝑐, 𝑑}))
78 dfsbcq 3722 . . . . . . . . . . 11 (𝑞 = {𝑐, 𝑑} → ([𝑞 / 𝑝]𝜓[{𝑐, 𝑑} / 𝑝]𝜓))
79 eqeq2 2810 . . . . . . . . . . 11 (𝑞 = {𝑐, 𝑑} → ({𝑎, 𝑏} = 𝑞 ↔ {𝑎, 𝑏} = {𝑐, 𝑑}))
8078, 79imbi12d 348 . . . . . . . . . 10 (𝑞 = {𝑐, 𝑑} → (([𝑞 / 𝑝]𝜓 → {𝑎, 𝑏} = 𝑞) ↔ ([{𝑐, 𝑑} / 𝑝]𝜓 → {𝑎, 𝑏} = {𝑐, 𝑑})))
8177, 80syl5ibrcom 250 . . . . . . . . 9 ((((𝑋𝑉 ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝜒 ∧ ∀𝑥𝑋𝑦𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))) ∧ (𝑐𝑋𝑑𝑋)) → (𝑞 = {𝑐, 𝑑} → ([𝑞 / 𝑝]𝜓 → {𝑎, 𝑏} = 𝑞)))
8281rexlimdvva 3253 . . . . . . . 8 (((𝑋𝑉 ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝜒 ∧ ∀𝑥𝑋𝑦𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))) → (∃𝑐𝑋𝑑𝑋 𝑞 = {𝑐, 𝑑} → ([𝑞 / 𝑝]𝜓 → {𝑎, 𝑏} = 𝑞)))
83 sprel 44016 . . . . . . . 8 (𝑞 ∈ (Pairs‘𝑋) → ∃𝑐𝑋𝑑𝑋 𝑞 = {𝑐, 𝑑})
8482, 83impel 509 . . . . . . 7 ((((𝑋𝑉 ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝜒 ∧ ∀𝑥𝑋𝑦𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))) ∧ 𝑞 ∈ (Pairs‘𝑋)) → ([𝑞 / 𝑝]𝜓 → {𝑎, 𝑏} = 𝑞))
8584ralrimiva 3149 . . . . . 6 (((𝑋𝑉 ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝜒 ∧ ∀𝑥𝑋𝑦𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))) → ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → {𝑎, 𝑏} = 𝑞))
86 nfv 1915 . . . . . . . 8 𝑝𝜒
87 nfcv 2955 . . . . . . . . 9 𝑝(Pairs‘𝑋)
88 nfv 1915 . . . . . . . . . 10 𝑝{𝑎, 𝑏} = 𝑞
891, 88nfim 1897 . . . . . . . . 9 𝑝([𝑞 / 𝑝]𝜓 → {𝑎, 𝑏} = 𝑞)
9087, 89nfralw 3189 . . . . . . . 8 𝑝𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → {𝑎, 𝑏} = 𝑞)
9186, 90nfan 1900 . . . . . . 7 𝑝(𝜒 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → {𝑎, 𝑏} = 𝑞))
92 eqeq1 2802 . . . . . . . . . 10 (𝑝 = {𝑎, 𝑏} → (𝑝 = 𝑞 ↔ {𝑎, 𝑏} = 𝑞))
9392imbi2d 344 . . . . . . . . 9 (𝑝 = {𝑎, 𝑏} → (([𝑞 / 𝑝]𝜓𝑝 = 𝑞) ↔ ([𝑞 / 𝑝]𝜓 → {𝑎, 𝑏} = 𝑞)))
9493ralbidv 3162 . . . . . . . 8 (𝑝 = {𝑎, 𝑏} → (∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓𝑝 = 𝑞) ↔ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → {𝑎, 𝑏} = 𝑞)))
957, 94anbi12d 633 . . . . . . 7 (𝑝 = {𝑎, 𝑏} → ((𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓𝑝 = 𝑞)) ↔ (𝜒 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → {𝑎, 𝑏} = 𝑞))))
9691, 95rspce 3560 . . . . . 6 (({𝑎, 𝑏} ∈ (Pairs‘𝑋) ∧ (𝜒 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓 → {𝑎, 𝑏} = 𝑞))) → ∃𝑝 ∈ (Pairs‘𝑋)(𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓𝑝 = 𝑞)))
9749, 50, 85, 96syl12anc 835 . . . . 5 (((𝑋𝑉 ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝜒 ∧ ∀𝑥𝑋𝑦𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))) → ∃𝑝 ∈ (Pairs‘𝑋)(𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓𝑝 = 𝑞)))
9897ex 416 . . . 4 ((𝑋𝑉 ∧ (𝑎𝑋𝑏𝑋)) → ((𝜒 ∧ ∀𝑥𝑋𝑦𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏})) → ∃𝑝 ∈ (Pairs‘𝑋)(𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓𝑝 = 𝑞))))
9998rexlimdvva 3253 . . 3 (𝑋𝑉 → (∃𝑎𝑋𝑏𝑋 (𝜒 ∧ ∀𝑥𝑋𝑦𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏})) → ∃𝑝 ∈ (Pairs‘𝑋)(𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓𝑝 = 𝑞))))
10047, 99impbid 215 . 2 (𝑋𝑉 → (∃𝑝 ∈ (Pairs‘𝑋)(𝜓 ∧ ∀𝑞 ∈ (Pairs‘𝑋)([𝑞 / 𝑝]𝜓𝑝 = 𝑞)) ↔ ∃𝑎𝑋𝑏𝑋 (𝜒 ∧ ∀𝑥𝑋𝑦𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))))
1015, 100syl5bb 286 1 (𝑋𝑉 → (∃!𝑝 ∈ (Pairs‘𝑋)𝜓 ↔ ∃𝑎𝑋𝑏𝑋 (𝜒 ∧ ∀𝑥𝑋𝑦𝑋 (𝜃 → {𝑥, 𝑦} = {𝑎, 𝑏}))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  ∃!wreu 3108  [wsbc 3720  {cpr 4527  ‘cfv 6324  Pairscspr 44009 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-spr 44010 This theorem is referenced by:  reuprpr  44055  reuopreuprim  44058
 Copyright terms: Public domain W3C validator