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Theorem opprb 46413
Description: Equality for unordered pairs corresponds to equality of unordered pairs with the same elements. (Contributed by AV, 9-Jul-2023.)
Assertion
Ref Expression
opprb (((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑋𝐷𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ∨ ⟨𝐴, 𝐵⟩ = ⟨𝐷, 𝐶⟩)))

Proof of Theorem opprb
StepHypRef Expression
1 preq12bg 4855 . 2 (((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑋𝐷𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
2 opthg 5479 . . . 4 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
32adantr 480 . . 3 (((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑋𝐷𝑌)) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
4 opthg 5479 . . . 4 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐷, 𝐶⟩ ↔ (𝐴 = 𝐷𝐵 = 𝐶)))
54adantr 480 . . 3 (((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑋𝐷𝑌)) → (⟨𝐴, 𝐵⟩ = ⟨𝐷, 𝐶⟩ ↔ (𝐴 = 𝐷𝐵 = 𝐶)))
63, 5orbi12d 917 . 2 (((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑋𝐷𝑌)) → ((⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ∨ ⟨𝐴, 𝐵⟩ = ⟨𝐷, 𝐶⟩) ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
71, 6bitr4d 282 1 (((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑋𝐷𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ∨ ⟨𝐴, 𝐵⟩ = ⟨𝐷, 𝐶⟩)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wo 846   = wceq 1534  wcel 2099  {cpr 4631  cop 4635
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-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
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-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636
This theorem is referenced by: (None)
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