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Theorem preq12i 4056
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 4053 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  { A ,  C }  =  { B ,  D } )
41, 2, 3mp2an 670 1  |-  { A ,  C }  =  { B ,  D }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1405   {cpr 3974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-v 3061  df-un 3419  df-sn 3973  df-pr 3975
This theorem is referenced by:  grpbasex  14956  grpplusgx  14957  indistpsx  19803  lgsdir2lem5  23983  wlkntrllem2  24979  clwwlkgt0  25188  tgrpset  33764  zlmodzxzadd  38458  zlmodzxzequa  38608  zlmodzxzequap  38611
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