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Theorem preq12i 4737
Description: Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
Hypotheses
Ref Expression
preq1i.1 𝐴 = 𝐵
preq12i.2 𝐶 = 𝐷
Assertion
Ref Expression
preq12i {𝐴, 𝐶} = {𝐵, 𝐷}

Proof of Theorem preq12i
StepHypRef Expression
1 preq1i.1 . 2 𝐴 = 𝐵
2 preq12i.2 . 2 𝐶 = 𝐷
3 preq12 4734 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → {𝐴, 𝐶} = {𝐵, 𝐷})
41, 2, 3mp2an 689 1 {𝐴, 𝐶} = {𝐵, 𝐷}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  {cpr 4625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-un 3948  df-sn 4624  df-pr 4626
This theorem is referenced by:  grpbasex  17243  grpplusgx  17244  indistpsx  22864  lgsdir2lem5  27213  wlk2v2elem2  29914  tgrpset  40127  zlmodzxzadd  47291  zlmodzxzequa  47433  zlmodzxzequap  47436
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