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Theorem oppr 44227
Description: Equality for ordered pairs implies equality of unordered pairs with the same elements. (Contributed by AV, 9-Jul-2023.)
Assertion
Ref Expression
oppr ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {𝐴, 𝐵} = {𝐶, 𝐷}))

Proof of Theorem oppr
StepHypRef Expression
1 opthg 5376 . 2 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
2 preq12 4666 . 2 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})
31, 2syl6bi 256 1 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {𝐴, 𝐵} = {𝐶, 𝐷}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2111  {cpr 4558  cop 4562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-ext 2709  ax-sep 5207  ax-nul 5214  ax-pr 5337
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3071  df-v 3423  df-dif 3884  df-un 3886  df-nul 4253  df-if 4455  df-sn 4557  df-pr 4559  df-op 4563
This theorem is referenced by:  or2expropbi  44231  ich2exprop  44627  2exopprim  44681  reuopreuprim  44682
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