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Theorem opreu2reu1 32275
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 32270 . 2 (∃!𝑥𝐴 , 𝑦𝐵𝜑 ↔ (∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑))
2 opreu2reu1.a . . 3 (𝑝 = ⟨𝑥, 𝑦⟩ → (𝜒𝜑))
32opreu2reurex 6292 . 2 (∃!𝑝 ∈ (𝐴 × 𝐵)𝜒 ↔ (∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑))
41, 3bitr4i 278 1 (∃!𝑥𝐴 , 𝑦𝐵𝜑 ↔ ∃!𝑝 ∈ (𝐴 × 𝐵)𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  wrex 3066  ∃!wreu 3370  cop 4630   × cxp 5670  ∃!w2reu 32269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-iun 4993  df-opab 5205  df-xp 5678  df-rel 5679  df-2reu 32270
This theorem is referenced by: (None)
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