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Theorem opreu2reu1 32327
Description: Equivalent definition of the double restricted existential uniqueness quantifier, using uniqueness of ordered pairs. (Contributed by Thierry Arnoux, 4-Jul-2023.)
Hypothesis
Ref Expression
opreu2reu1.a (𝑝 = ⟨𝑥, 𝑦⟩ → (𝜒𝜑))
Assertion
Ref Expression
opreu2reu1 (∃!𝑥𝐴 , 𝑦𝐵𝜑 ↔ ∃!𝑝 ∈ (𝐴 × 𝐵)𝜒)
Distinct variable groups:   𝐴,𝑝,𝑥,𝑦   𝐵,𝑝,𝑥,𝑦   𝜒,𝑥,𝑦   𝜑,𝑝,𝑥,𝑦
Allowed substitution hint:   𝜒(𝑝)

Proof of Theorem opreu2reu1
StepHypRef Expression
1 df-2reu 32322 . 2 (∃!𝑥𝐴 , 𝑦𝐵𝜑 ↔ (∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑))
2 opreu2reu1.a . . 3 (𝑝 = ⟨𝑥, 𝑦⟩ → (𝜒𝜑))
32opreu2reurex 6293 . 2 (∃!𝑝 ∈ (𝐴 × 𝐵)𝜒 ↔ (∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑))
41, 3bitr4i 277 1 (∃!𝑥𝐴 , 𝑦𝐵𝜑 ↔ ∃!𝑝 ∈ (𝐴 × 𝐵)𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wrex 3060  ∃!wreu 3362  cop 4630   × cxp 5670  ∃!w2reu 32321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-iun 4993  df-opab 5206  df-xp 5678  df-rel 5679  df-2reu 32322
This theorem is referenced by: (None)
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