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Theorem preqr1g 4738
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 4734. (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 486 . . 3 ((𝐴𝑉𝐵𝑊) → 𝐴𝑉)
2 simpr 488 . . 3 ((𝐴𝑉𝐵𝑊) → 𝐵𝑊)
31, 2preq1b 4732 . 2 ((𝐴𝑉𝐵𝑊) → ({𝐴, 𝐶} = {𝐵, 𝐶} ↔ 𝐴 = 𝐵))
43biimpd 232 1 ((𝐴𝑉𝐵𝑊) → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2114  {cpr 4518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-v 3400  df-un 3848  df-sn 4517  df-pr 4519
This theorem is referenced by:  umgr2adedgspth  27886
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