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Theorem preqr1g 4783
Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. Closed form of preqr1 4779. (Contributed by AV, 29-Jan-2021.) (Revised by AV, 18-Sep-2021.)
Assertion
Ref Expression
preqr1g ((𝐴𝑉𝐵𝑊) → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵))

Proof of Theorem preqr1g
StepHypRef Expression
1 simpl 483 . . 3 ((𝐴𝑉𝐵𝑊) → 𝐴𝑉)
2 simpr 485 . . 3 ((𝐴𝑉𝐵𝑊) → 𝐵𝑊)
31, 2preq1b 4777 . 2 ((𝐴𝑉𝐵𝑊) → ({𝐴, 𝐶} = {𝐵, 𝐶} ↔ 𝐴 = 𝐵))
43biimpd 228 1 ((𝐴𝑉𝐵𝑊) → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  {cpr 4563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-un 3892  df-sn 4562  df-pr 4564
This theorem is referenced by:  umgr2adedgspth  28313
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