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Theorem reu3op 6290
Description: There is a unique ordered pair fulfilling a wff iff there are uniquely two sets fulfilling a corresponding wff. (Contributed by AV, 1-Jul-2023.)
Hypothesis
Ref Expression
reu3op.a (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜓𝜒))
Assertion
Ref Expression
reu3op (∃!𝑝 ∈ (𝑋 × 𝑌)𝜓 ↔ (∃𝑎𝑋𝑏𝑌 𝜒 ∧ ∃𝑥𝑋𝑦𝑌𝑎𝑋𝑏𝑌 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)))
Distinct variable groups:   𝑋,𝑎,𝑏,𝑝,𝑥,𝑦   𝑌,𝑎,𝑏,𝑝,𝑥,𝑦   𝜓,𝑎,𝑏,𝑥,𝑦   𝜒,𝑝
Allowed substitution hints:   𝜓(𝑝)   𝜒(𝑥,𝑦,𝑎,𝑏)

Proof of Theorem reu3op
Dummy variable 𝑞 is distinct from all other variables.
StepHypRef Expression
1 reu3 3720 . 2 (∃!𝑝 ∈ (𝑋 × 𝑌)𝜓 ↔ (∃𝑝 ∈ (𝑋 × 𝑌)𝜓 ∧ ∃𝑞 ∈ (𝑋 × 𝑌)∀𝑝 ∈ (𝑋 × 𝑌)(𝜓𝑝 = 𝑞)))
2 reu3op.a . . . 4 (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜓𝜒))
32rexxp 5839 . . 3 (∃𝑝 ∈ (𝑋 × 𝑌)𝜓 ↔ ∃𝑎𝑋𝑏𝑌 𝜒)
4 eqeq2 2739 . . . . . . 7 (𝑞 = ⟨𝑥, 𝑦⟩ → (𝑝 = 𝑞𝑝 = ⟨𝑥, 𝑦⟩))
54imbi2d 340 . . . . . 6 (𝑞 = ⟨𝑥, 𝑦⟩ → ((𝜓𝑝 = 𝑞) ↔ (𝜓𝑝 = ⟨𝑥, 𝑦⟩)))
65ralbidv 3172 . . . . 5 (𝑞 = ⟨𝑥, 𝑦⟩ → (∀𝑝 ∈ (𝑋 × 𝑌)(𝜓𝑝 = 𝑞) ↔ ∀𝑝 ∈ (𝑋 × 𝑌)(𝜓𝑝 = ⟨𝑥, 𝑦⟩)))
76rexxp 5839 . . . 4 (∃𝑞 ∈ (𝑋 × 𝑌)∀𝑝 ∈ (𝑋 × 𝑌)(𝜓𝑝 = 𝑞) ↔ ∃𝑥𝑋𝑦𝑌𝑝 ∈ (𝑋 × 𝑌)(𝜓𝑝 = ⟨𝑥, 𝑦⟩))
8 eqeq1 2731 . . . . . . . 8 (𝑝 = ⟨𝑎, 𝑏⟩ → (𝑝 = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩))
92, 8imbi12d 344 . . . . . . 7 (𝑝 = ⟨𝑎, 𝑏⟩ → ((𝜓𝑝 = ⟨𝑥, 𝑦⟩) ↔ (𝜒 → ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩)))
109ralxp 5838 . . . . . 6 (∀𝑝 ∈ (𝑋 × 𝑌)(𝜓𝑝 = ⟨𝑥, 𝑦⟩) ↔ ∀𝑎𝑋𝑏𝑌 (𝜒 → ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩))
11 eqcom 2734 . . . . . . . . 9 (⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)
1211a1i 11 . . . . . . . 8 (((𝑥𝑋𝑦𝑌) ∧ (𝑎𝑋𝑏𝑌)) → (⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩))
1312imbi2d 340 . . . . . . 7 (((𝑥𝑋𝑦𝑌) ∧ (𝑎𝑋𝑏𝑌)) → ((𝜒 → ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩) ↔ (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)))
14132ralbidva 3211 . . . . . 6 ((𝑥𝑋𝑦𝑌) → (∀𝑎𝑋𝑏𝑌 (𝜒 → ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩) ↔ ∀𝑎𝑋𝑏𝑌 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)))
1510, 14bitrid 283 . . . . 5 ((𝑥𝑋𝑦𝑌) → (∀𝑝 ∈ (𝑋 × 𝑌)(𝜓𝑝 = ⟨𝑥, 𝑦⟩) ↔ ∀𝑎𝑋𝑏𝑌 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)))
16152rexbiia 3210 . . . 4 (∃𝑥𝑋𝑦𝑌𝑝 ∈ (𝑋 × 𝑌)(𝜓𝑝 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑥𝑋𝑦𝑌𝑎𝑋𝑏𝑌 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩))
177, 16bitri 275 . . 3 (∃𝑞 ∈ (𝑋 × 𝑌)∀𝑝 ∈ (𝑋 × 𝑌)(𝜓𝑝 = 𝑞) ↔ ∃𝑥𝑋𝑦𝑌𝑎𝑋𝑏𝑌 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩))
183, 17anbi12i 626 . 2 ((∃𝑝 ∈ (𝑋 × 𝑌)𝜓 ∧ ∃𝑞 ∈ (𝑋 × 𝑌)∀𝑝 ∈ (𝑋 × 𝑌)(𝜓𝑝 = 𝑞)) ↔ (∃𝑎𝑋𝑏𝑌 𝜒 ∧ ∃𝑥𝑋𝑦𝑌𝑎𝑋𝑏𝑌 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)))
191, 18bitri 275 1 (∃!𝑝 ∈ (𝑋 × 𝑌)𝜓 ↔ (∃𝑎𝑋𝑏𝑌 𝜒 ∧ ∃𝑥𝑋𝑦𝑌𝑎𝑋𝑏𝑌 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  wcel 2099  wral 3056  wrex 3065  ∃!wreu 3369  cop 4630   × cxp 5670
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 2164  ax-ext 2698  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 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  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
This theorem is referenced by:  opreu2reurex  6292
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