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Theorem opreu2reurex 6023
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 2801 . . . . . . . 8 (⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ↔ ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩)
2 vex 3439 . . . . . . . . 9 𝑎 ∈ V
3 vex 3439 . . . . . . . . 9 𝑏 ∈ V
42, 3opth 5263 . . . . . . . 8 (⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑎 = 𝑥𝑏 = 𝑦))
51, 4bitri 276 . . . . . . 7 (⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ↔ (𝑎 = 𝑥𝑏 = 𝑦))
65imbi2i 337 . . . . . 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 3164 . . . 4 ((𝑥𝐴𝑦𝐵) → (∀𝑎𝐴𝑏𝐵 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩) ↔ ∀𝑎𝐴𝑏𝐵 (𝜒 → (𝑎 = 𝑥𝑏 = 𝑦))))
982rexbiia 3260 . . 3 (∃𝑥𝐴𝑦𝐵𝑎𝐴𝑏𝐵 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩) ↔ ∃𝑥𝐴𝑦𝐵𝑎𝐴𝑏𝐵 (𝜒 → (𝑎 = 𝑥𝑏 = 𝑦)))
109anbi2i 622 . 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 6021 . 2 (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 ↔ (∃𝑎𝐴𝑏𝐵 𝜒 ∧ ∃𝑥𝐴𝑦𝐵𝑎𝐴𝑏𝐵 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)))
13 2reu4 4382 . 2 ((∃!𝑎𝐴𝑏𝐵 𝜒 ∧ ∃!𝑏𝐵𝑎𝐴 𝜒) ↔ (∃𝑎𝐴𝑏𝐵 𝜒 ∧ ∃𝑥𝐴𝑦𝐵𝑎𝐴𝑏𝐵 (𝜒 → (𝑎 = 𝑥𝑏 = 𝑦))))
1410, 12, 133bitr4i 304 1 (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 ↔ (∃!𝑎𝐴𝑏𝐵 𝜒 ∧ ∃!𝑏𝐵𝑎𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1522  wcel 2080  wral 3104  wrex 3105  ∃!wreu 3106  cop 4480   × cxp 5444
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-sep 5097  ax-nul 5104  ax-pr 5224
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-rmo 3112  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-sn 4475  df-pr 4477  df-op 4481  df-iun 4829  df-opab 5027  df-xp 5452  df-rel 5453
This theorem is referenced by:  opreu2reu  6024  2sqreuop  25720  2sqreuopnn  25721  2sqreuoplt  25722  2sqreuopltb  25723  2sqreuopnnlt  25724  2sqreuopnnltb  25725  opreu2reu1  29931
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