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Theorem reu3op 6111
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 3666 . 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 5677 . . 3 (∃𝑝 ∈ (𝑋 × 𝑌)𝜓 ↔ ∃𝑎𝑋𝑏𝑌 𝜒)
4 eqeq2 2810 . . . . . . 7 (𝑞 = ⟨𝑥, 𝑦⟩ → (𝑝 = 𝑞𝑝 = ⟨𝑥, 𝑦⟩))
54imbi2d 344 . . . . . 6 (𝑞 = ⟨𝑥, 𝑦⟩ → ((𝜓𝑝 = 𝑞) ↔ (𝜓𝑝 = ⟨𝑥, 𝑦⟩)))
65ralbidv 3162 . . . . 5 (𝑞 = ⟨𝑥, 𝑦⟩ → (∀𝑝 ∈ (𝑋 × 𝑌)(𝜓𝑝 = 𝑞) ↔ ∀𝑝 ∈ (𝑋 × 𝑌)(𝜓𝑝 = ⟨𝑥, 𝑦⟩)))
76rexxp 5677 . . . 4 (∃𝑞 ∈ (𝑋 × 𝑌)∀𝑝 ∈ (𝑋 × 𝑌)(𝜓𝑝 = 𝑞) ↔ ∃𝑥𝑋𝑦𝑌𝑝 ∈ (𝑋 × 𝑌)(𝜓𝑝 = ⟨𝑥, 𝑦⟩))
8 eqeq1 2802 . . . . . . . 8 (𝑝 = ⟨𝑎, 𝑏⟩ → (𝑝 = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩))
92, 8imbi12d 348 . . . . . . 7 (𝑝 = ⟨𝑎, 𝑏⟩ → ((𝜓𝑝 = ⟨𝑥, 𝑦⟩) ↔ (𝜒 → ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩)))
109ralxp 5676 . . . . . 6 (∀𝑝 ∈ (𝑋 × 𝑌)(𝜓𝑝 = ⟨𝑥, 𝑦⟩) ↔ ∀𝑎𝑋𝑏𝑌 (𝜒 → ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩))
11 eqcom 2805 . . . . . . . . 9 (⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)
1211a1i 11 . . . . . . . 8 (((𝑥𝑋𝑦𝑌) ∧ (𝑎𝑋𝑏𝑌)) → (⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩))
1312imbi2d 344 . . . . . . 7 (((𝑥𝑋𝑦𝑌) ∧ (𝑎𝑋𝑏𝑌)) → ((𝜒 → ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩) ↔ (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)))
14132ralbidva 3163 . . . . . 6 ((𝑥𝑋𝑦𝑌) → (∀𝑎𝑋𝑏𝑌 (𝜒 → ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩) ↔ ∀𝑎𝑋𝑏𝑌 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)))
1510, 14syl5bb 286 . . . . 5 ((𝑥𝑋𝑦𝑌) → (∀𝑝 ∈ (𝑋 × 𝑌)(𝜓𝑝 = ⟨𝑥, 𝑦⟩) ↔ ∀𝑎𝑋𝑏𝑌 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)))
16152rexbiia 3257 . . . 4 (∃𝑥𝑋𝑦𝑌𝑝 ∈ (𝑋 × 𝑌)(𝜓𝑝 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑥𝑋𝑦𝑌𝑎𝑋𝑏𝑌 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩))
177, 16bitri 278 . . 3 (∃𝑞 ∈ (𝑋 × 𝑌)∀𝑝 ∈ (𝑋 × 𝑌)(𝜓𝑝 = 𝑞) ↔ ∃𝑥𝑋𝑦𝑌𝑎𝑋𝑏𝑌 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩))
183, 17anbi12i 629 . 2 ((∃𝑝 ∈ (𝑋 × 𝑌)𝜓 ∧ ∃𝑞 ∈ (𝑋 × 𝑌)∀𝑝 ∈ (𝑋 × 𝑌)(𝜓𝑝 = 𝑞)) ↔ (∃𝑎𝑋𝑏𝑌 𝜒 ∧ ∃𝑥𝑋𝑦𝑌𝑎𝑋𝑏𝑌 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)))
191, 18bitri 278 1 (∃!𝑝 ∈ (𝑋 × 𝑌)𝜓 ↔ (∃𝑎𝑋𝑏𝑌 𝜒 ∧ ∃𝑥𝑋𝑦𝑌𝑎𝑋𝑏𝑌 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2111  wral 3106  wrex 3107  ∃!wreu 3108  cop 4531   × cxp 5517
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-iun 4883  df-opab 5093  df-xp 5525  df-rel 5526
This theorem is referenced by:  opreu2reurex  6113
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