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Theorem opreu2reurex 6314
Description: There is a unique ordered pair fulfilling a wff iff there are uniquely two sets fulfilling a corresponding wff. (Contributed by AV, 24-Jun-2023.) (Revised by AV, 1-Jul-2023.)
Hypothesis
Ref Expression
opreu2reurex.a (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑𝜒))
Assertion
Ref Expression
opreu2reurex (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 ↔ (∃!𝑎𝐴𝑏𝐵 𝜒 ∧ ∃!𝑏𝐵𝑎𝐴 𝜒))
Distinct variable groups:   𝐴,𝑎,𝑏,𝑝   𝐵,𝑎,𝑏,𝑝   𝜑,𝑎,𝑏   𝜒,𝑝
Allowed substitution hints:   𝜑(𝑝)   𝜒(𝑎,𝑏)

Proof of Theorem opreu2reurex
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqcom 2744 . . . . . . . 8 (⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ↔ ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩)
2 vex 3484 . . . . . . . . 9 𝑎 ∈ V
3 vex 3484 . . . . . . . . 9 𝑏 ∈ V
42, 3opth 5481 . . . . . . . 8 (⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑎 = 𝑥𝑏 = 𝑦))
51, 4bitri 275 . . . . . . 7 (⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ↔ (𝑎 = 𝑥𝑏 = 𝑦))
65imbi2i 336 . . . . . 6 ((𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩) ↔ (𝜒 → (𝑎 = 𝑥𝑏 = 𝑦)))
76a1i 11 . . . . 5 (((𝑥𝐴𝑦𝐵) ∧ (𝑎𝐴𝑏𝐵)) → ((𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩) ↔ (𝜒 → (𝑎 = 𝑥𝑏 = 𝑦))))
872ralbidva 3219 . . . 4 ((𝑥𝐴𝑦𝐵) → (∀𝑎𝐴𝑏𝐵 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩) ↔ ∀𝑎𝐴𝑏𝐵 (𝜒 → (𝑎 = 𝑥𝑏 = 𝑦))))
982rexbiia 3218 . . 3 (∃𝑥𝐴𝑦𝐵𝑎𝐴𝑏𝐵 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩) ↔ ∃𝑥𝐴𝑦𝐵𝑎𝐴𝑏𝐵 (𝜒 → (𝑎 = 𝑥𝑏 = 𝑦)))
109anbi2i 623 . 2 ((∃𝑎𝐴𝑏𝐵 𝜒 ∧ ∃𝑥𝐴𝑦𝐵𝑎𝐴𝑏𝐵 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)) ↔ (∃𝑎𝐴𝑏𝐵 𝜒 ∧ ∃𝑥𝐴𝑦𝐵𝑎𝐴𝑏𝐵 (𝜒 → (𝑎 = 𝑥𝑏 = 𝑦))))
11 opreu2reurex.a . . 3 (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑𝜒))
1211reu3op 6312 . 2 (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 ↔ (∃𝑎𝐴𝑏𝐵 𝜒 ∧ ∃𝑥𝐴𝑦𝐵𝑎𝐴𝑏𝐵 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)))
13 2reu4 4523 . 2 ((∃!𝑎𝐴𝑏𝐵 𝜒 ∧ ∃!𝑏𝐵𝑎𝐴 𝜒) ↔ (∃𝑎𝐴𝑏𝐵 𝜒 ∧ ∃𝑥𝐴𝑦𝐵𝑎𝐴𝑏𝐵 (𝜒 → (𝑎 = 𝑥𝑏 = 𝑦))))
1410, 12, 133bitr4i 303 1 (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 ↔ (∃!𝑎𝐴𝑏𝐵 𝜒 ∧ ∃!𝑏𝐵𝑎𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  wrex 3070  ∃!wreu 3378  cop 4632   × cxp 5683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-iun 4993  df-opab 5206  df-xp 5691  df-rel 5692
This theorem is referenced by:  opreu2reu  6315  2sqreuop  27506  2sqreuopnn  27507  2sqreuoplt  27508  2sqreuopltb  27509  2sqreuopnnlt  27510  2sqreuopnnltb  27511  opreu2reu1  32503
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