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Theorem preqr1g 4849
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 4845. (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 482 . . 3 ((𝐴𝑉𝐵𝑊) → 𝐴𝑉)
2 simpr 484 . . 3 ((𝐴𝑉𝐵𝑊) → 𝐵𝑊)
31, 2preq1b 4843 . 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 395   = wceq 1534  wcel 2099  {cpr 4626
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
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3472  df-un 3950  df-sn 4625  df-pr 4627
This theorem is referenced by:  umgr2adedgspth  29752
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