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Theorem opreu2reurex 6290
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 2739 . . . . . . . 8 (⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ↔ ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩)
2 vex 3478 . . . . . . . . 9 𝑎 ∈ V
3 vex 3478 . . . . . . . . 9 𝑏 ∈ V
42, 3opth 5475 . . . . . . . 8 (⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑎 = 𝑥𝑏 = 𝑦))
51, 4bitri 274 . . . . . . 7 (⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ↔ (𝑎 = 𝑥𝑏 = 𝑦))
65imbi2i 335 . . . . . 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 3216 . . . 4 ((𝑥𝐴𝑦𝐵) → (∀𝑎𝐴𝑏𝐵 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩) ↔ ∀𝑎𝐴𝑏𝐵 (𝜒 → (𝑎 = 𝑥𝑏 = 𝑦))))
982rexbiia 3215 . . 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 6288 . 2 (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 ↔ (∃𝑎𝐴𝑏𝐵 𝜒 ∧ ∃𝑥𝐴𝑦𝐵𝑎𝐴𝑏𝐵 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)))
13 2reu4 4525 . 2 ((∃!𝑎𝐴𝑏𝐵 𝜒 ∧ ∃!𝑏𝐵𝑎𝐴 𝜒) ↔ (∃𝑎𝐴𝑏𝐵 𝜒 ∧ ∃𝑥𝐴𝑦𝐵𝑎𝐴𝑏𝐵 (𝜒 → (𝑎 = 𝑥𝑏 = 𝑦))))
1410, 12, 133bitr4i 302 1 (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 ↔ (∃!𝑎𝐴𝑏𝐵 𝜒 ∧ ∃!𝑏𝐵𝑎𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3061  wrex 3070  ∃!wreu 3374  cop 4633   × cxp 5673
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-iun 4998  df-opab 5210  df-xp 5681  df-rel 5682
This theorem is referenced by:  opreu2reu  6291  2sqreuop  26954  2sqreuopnn  26955  2sqreuoplt  26956  2sqreuopltb  26957  2sqreuopnnlt  26958  2sqreuopnnltb  26959  opreu2reu1  31711
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