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Theorem reu3op 6194
Description: There is a unique ordered pair fulfilling a wff iff there are uniquely two sets fulfilling a corresponding wff. (Contributed by AV, 1-Jul-2023.)
Hypothesis
Ref Expression
reu3op.a (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜓𝜒))
Assertion
Ref Expression
reu3op (∃!𝑝 ∈ (𝑋 × 𝑌)𝜓 ↔ (∃𝑎𝑋𝑏𝑌 𝜒 ∧ ∃𝑥𝑋𝑦𝑌𝑎𝑋𝑏𝑌 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)))
Distinct variable groups:   𝑋,𝑎,𝑏,𝑝,𝑥,𝑦   𝑌,𝑎,𝑏,𝑝,𝑥,𝑦   𝜓,𝑎,𝑏,𝑥,𝑦   𝜒,𝑝
Allowed substitution hints:   𝜓(𝑝)   𝜒(𝑥,𝑦,𝑎,𝑏)

Proof of Theorem reu3op
Dummy variable 𝑞 is distinct from all other variables.
StepHypRef Expression
1 reu3 3666 . 2 (∃!𝑝 ∈ (𝑋 × 𝑌)𝜓 ↔ (∃𝑝 ∈ (𝑋 × 𝑌)𝜓 ∧ ∃𝑞 ∈ (𝑋 × 𝑌)∀𝑝 ∈ (𝑋 × 𝑌)(𝜓𝑝 = 𝑞)))
2 reu3op.a . . . 4 (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜓𝜒))
32rexxp 5750 . . 3 (∃𝑝 ∈ (𝑋 × 𝑌)𝜓 ↔ ∃𝑎𝑋𝑏𝑌 𝜒)
4 eqeq2 2752 . . . . . . 7 (𝑞 = ⟨𝑥, 𝑦⟩ → (𝑝 = 𝑞𝑝 = ⟨𝑥, 𝑦⟩))
54imbi2d 341 . . . . . 6 (𝑞 = ⟨𝑥, 𝑦⟩ → ((𝜓𝑝 = 𝑞) ↔ (𝜓𝑝 = ⟨𝑥, 𝑦⟩)))
65ralbidv 3123 . . . . 5 (𝑞 = ⟨𝑥, 𝑦⟩ → (∀𝑝 ∈ (𝑋 × 𝑌)(𝜓𝑝 = 𝑞) ↔ ∀𝑝 ∈ (𝑋 × 𝑌)(𝜓𝑝 = ⟨𝑥, 𝑦⟩)))
76rexxp 5750 . . . 4 (∃𝑞 ∈ (𝑋 × 𝑌)∀𝑝 ∈ (𝑋 × 𝑌)(𝜓𝑝 = 𝑞) ↔ ∃𝑥𝑋𝑦𝑌𝑝 ∈ (𝑋 × 𝑌)(𝜓𝑝 = ⟨𝑥, 𝑦⟩))
8 eqeq1 2744 . . . . . . . 8 (𝑝 = ⟨𝑎, 𝑏⟩ → (𝑝 = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩))
92, 8imbi12d 345 . . . . . . 7 (𝑝 = ⟨𝑎, 𝑏⟩ → ((𝜓𝑝 = ⟨𝑥, 𝑦⟩) ↔ (𝜒 → ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩)))
109ralxp 5749 . . . . . 6 (∀𝑝 ∈ (𝑋 × 𝑌)(𝜓𝑝 = ⟨𝑥, 𝑦⟩) ↔ ∀𝑎𝑋𝑏𝑌 (𝜒 → ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩))
11 eqcom 2747 . . . . . . . . 9 (⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)
1211a1i 11 . . . . . . . 8 (((𝑥𝑋𝑦𝑌) ∧ (𝑎𝑋𝑏𝑌)) → (⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩))
1312imbi2d 341 . . . . . . 7 (((𝑥𝑋𝑦𝑌) ∧ (𝑎𝑋𝑏𝑌)) → ((𝜒 → ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩) ↔ (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)))
14132ralbidva 3124 . . . . . 6 ((𝑥𝑋𝑦𝑌) → (∀𝑎𝑋𝑏𝑌 (𝜒 → ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩) ↔ ∀𝑎𝑋𝑏𝑌 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)))
1510, 14bitrid 282 . . . . 5 ((𝑥𝑋𝑦𝑌) → (∀𝑝 ∈ (𝑋 × 𝑌)(𝜓𝑝 = ⟨𝑥, 𝑦⟩) ↔ ∀𝑎𝑋𝑏𝑌 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)))
16152rexbiia 3229 . . . 4 (∃𝑥𝑋𝑦𝑌𝑝 ∈ (𝑋 × 𝑌)(𝜓𝑝 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑥𝑋𝑦𝑌𝑎𝑋𝑏𝑌 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩))
177, 16bitri 274 . . 3 (∃𝑞 ∈ (𝑋 × 𝑌)∀𝑝 ∈ (𝑋 × 𝑌)(𝜓𝑝 = 𝑞) ↔ ∃𝑥𝑋𝑦𝑌𝑎𝑋𝑏𝑌 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩))
183, 17anbi12i 627 . 2 ((∃𝑝 ∈ (𝑋 × 𝑌)𝜓 ∧ ∃𝑞 ∈ (𝑋 × 𝑌)∀𝑝 ∈ (𝑋 × 𝑌)(𝜓𝑝 = 𝑞)) ↔ (∃𝑎𝑋𝑏𝑌 𝜒 ∧ ∃𝑥𝑋𝑦𝑌𝑎𝑋𝑏𝑌 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)))
191, 18bitri 274 1 (∃!𝑝 ∈ (𝑋 × 𝑌)𝜓 ↔ (∃𝑎𝑋𝑏𝑌 𝜒 ∧ ∃𝑥𝑋𝑦𝑌𝑎𝑋𝑏𝑌 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1542  wcel 2110  wral 3066  wrex 3067  ∃!wreu 3068  cop 4573   × cxp 5588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-iun 4932  df-opab 5142  df-xp 5596  df-rel 5597
This theorem is referenced by:  opreu2reurex  6196
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